1
KNOTS AND DISTRIBUTIVE HOMOLOGY:
arXiv:1409.7044v1 [math.GT] 24 Sep 2014
FROM ARC COLORINGS TO YANG-BAXTER HOMOLOGY
JÓZEF H. PRZYTYCKI
Contents
1. Introduction
1.1. Invariants of arc colorings
2. Monoid of binary operations
3. Homology of magmas
3.1. Homology of abstract simplicial complexes
3.2. Degenerate subcomplex
3.3. Bicomplex for a degenerate subcomplex CnD
3.4. Homology with coefficients in a k−Mod functor
4. Group homology of a semigroup
5. Hochschild homology of a semigroup and an algebra
6. Homology of distributive structures
6.1. One-term distributive homology
6.2. Multi-term distributive homology
7. Bloh-Leibniz-Loday algebra
8. Semigroup extensions and shelf extensions
8.1. Extensions in right distributive case
8.2. Extensions in entropic case
9. Degeneracy for a weak and very weak simplicial module
10. Degeneracy for a weak simplicial module
10.1. Right filtration of degenerate distributive elements
10.2. Integration maps ûi : F̂np → F̂np−1
10.3. Weak simplicial modules with integration
11. Degeneracy for a very weak simplicial module
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This paper has its roots in two series of talks I gave: in Russia (Lomonosov
Moscow State University, May 29-June 1, 2012), where the visualization in Figure
6.2 was observed, Korea (TAPU Workshop on Knot Theory, July 23-27, 2012),
and in a talk at Oberwolfach Conference (June 3-9, 2012). The short version of
this paper was published in Oberwolfach Proceedings [Prz-6]. While I keep novelty
of the talks (many new ideas were crystallized then), I added a lot of supporting
material so the paper is mostly self-sufficient. I also kept, to some extent, the
structure of talks; it may lead to some repetitions but I hope it is useful for a
reader.
1
Knots and distributive homology
2
11.1. Introduction to t-simplicial objects
12. From distributive homology to Yang-Baxter homology
12.1. Graphical visualization of Yang-Baxter face maps
13. Geometric realization of simplicial and cubic sets
13.1. Geometric realization of a (pre)cubic set
14. Higher dimensional knot theory M n → Rn+2
15. Acknowledgements
References
50
50
52
53
54
57
58
58
1. Introduction
While homology theory of associative structures, such as groups and
rings, has been extensively studied in the past beginning with the
work of Hurewicz, Hopf, Eilenberg, and Hochschild, the non-associative
structures, such as racks or quandles, were neglected until recently.
The distributive structures1 have been studied for a long time and
already C.S. Peirce (1839-1914) in 1880 [Peir] emphasized the importance of (right) self-distributivity in algebraic structures, and his friend
E. Schröder, [Schr] gave an example of a three element magma (X; ∗)
which is not associative2 (compare Section 2).
However, homology for such universal algebras was introduced only
between 1990 and 1995 by Fenn, Rourke, and Sanderson [FRS-1, FRS-2,
FRS-3, Fenn] . We develop theory in the historical context and propose
a general framework to study homology of distributive structures. We
outline potential relations to Khovanov homology and categorification,
via Yang-Baxter operators. We use here the fact that Yang-Baxter
equation can be thought of as a generalization of self-distributivity.
1.1. Invariants of arc colorings. Consider a link diagram D, say
, and a finite set X. We may define a diagram invariant to be the
1The
word distributivity was coined in 1814 by French mathematician Francois
Servois (1767-1847).
2The example Schröder (1841-1902) gave is
∗ 0 1 2
0 0 2 1
1 2 1 0
2 1 0 2
and we named elements of the magma by 0, 1 and 2 as this example is the base
for Fox three colorings of links (developed about 1956) [C-F, Cro, Prz-3], and the
operation can be written as x∗y = 2y −x modulo 3; it happen to be self-distributive
from both sides, that is (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z) and x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z)
(see Example 2.3(5)).
3
number of colorings of arcs3 of D by elements of X, colX (D). Even such
a naive definition leads to a link invariant colX (L) = minD∈L colX (D),
where D ∈ L means that D is a diagram of L.4 More sensible approach
would start with a magma (X; ∗), that is a set with binary operation,
and with the coloring convention of Figure 1.1.
b
a *b
b
a
Figure 1.1; convention for a magma coloring of a crossing
Again, let for a finite X, col(X;∗) (D) denote the number of colorings
of arcs of D by elements of X, according to the convention given in
Figure 1.1, at every crossing. We can define an oriented link invariant
by considering col(X;∗) (L) = minD∈L col(X;∗) (D); Alternatively, we can
minimize col(X;∗) (L) over minimal crossing diagrams of L only. Such
an invariant would be very difficult to compute so it is better to look
for properties of (X; ∗) so that col(X;∗) (D) is invariant under Reide) gives idempotent relation a ∗ a = a.5 R2
meister moves: R1 (
) forces ∗ to be an invertible operation, and the third move illus(
trated in detail in Figure 1.2 below, forces on ∗ a right self-distributivity
(a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c).
c
b
(a * c)
a
(a * c) * (b * c)
c
b*c
b
c
R
3
a
(a * b) * c
a *b
b*c
c
Figure 1.2; magma coloring of a third Reidemeister move and distributivity
3We
use the term arc for a part of the diagram from an undercrossing to the next
undercrossing (including possibility of a trivial component), and the term semi-arc
for a part of the diagram from a crossing to the next crossing. Thus, for example,
a standard trefoil knot diagram has three arcs and six semi-arcs.
4One can say that it is nonsense but an invariant is nontrivial: col (L) =
X
|X|cr(L)+t(L), where cr(L) is the crossing number of L and t(L) the number of
trivial components in L. This is the case as for a knot diagram D with at least one
crossing the number of arcs equals to the number of crossings.
5The names idempotent (same power) and nilpotent (zero power) were introduced
in 1870 by Benjamin Peirce [Pei], page 20.
Knots and distributive homology
4
The magma (X; ∗) satisfying all three conditions is called a quandle,
the last two – a rack, and only the last condition – a shelf or RDS
(right distributive system). Thus, if (X; ∗) is a quandle then col(X;∗) (D)
(which we denote from now on succinctly colX (D)) is a link invariant.
We also use the notation ColX (D) for the set of X-colorings of D,
thus colX (D) = |ColX (D)|. We also can try more generally to color
semi-arcs of a diagram by elements of X and declare for each colored
crossing a weight of the crossing. This approach leads to state sums
and Yang-Baxter operators (see Section 6 and Figures 6.2, 12.1, and
12.3).
We also can do more with distributive magmas (after S.Carter, S.Kamada,
and M.Saito [CKS]; compare also M.Greene thesis [Gr]). We can
sum over all crossings the pairs ±(a, b) according to the convention
a b
a b
b
b
P
; the investigation of invariance of
±(a, b) under
*
a
*
a
(a,b)
−(a,b)
Reidemeister moves was a hint toward construction of (co)homology of
quandles.
We also encounter right distributivity by asking the following question: for a given coloring φ ∈ ColX (D) by a magma (X; ∗), and an
element x ∈ X is coloring φ ∗ x also a magma coloring? If, as before, coloring is given by a, b, and c = a ∗ b then the new coloring
of a crossing is given by a ∗ x, b ∗ x, and c ∗ x. For a magma coloring we need (a ∗ b) ∗ x = (a ∗ x) ∗ (b ∗ x) which is exactly right
self-distributivity. To put it on a more solid footing, we observe that
the map ∗x : X → X with ∗x (a) = a ∗ x is a magma homomorphism:
∗x (a ∗ b) = (a ∗ b) ∗ x = (a ∗ x) ∗ (b ∗ x) = ∗x (a) ∗ ∗x (b). For any
magma homomorphism g : X → X if f : arcs(D) → X is a magma
coloring then gf defined by (gf )(arc) = g(f (arc)) is a magma coloring
of D. These are classical observations thus it is interesting to notice
the slightly more general fact concerning the following question:
Let (X; ∗1) be a magma and f and g two (X; ∗1 ) magma colorings
of a diagram D. Find the magma operation ∗2 so that f ∗2 g is also a
(X; ∗1) magma coloring. The question reduces to the previous one if
g is a trivial (i.e. constant) coloring and ∗1 = ∗2 . The nice condition
which answers the question was first discussed by M.Niebrzydowski at
his talk at Knots in Washington XXXV conference in December of 2012
[Nieb] (compare also [C-N]).
Lemma 1.1. Let f, g : arcs(D) → X be two colorings of a diagram
D by (X; ∗1) (that is f, g ∈ Col(X;∗1 ) (D)). Then f ∗2 g where ∗2 is
another binary operation on X and (f ∗2 g)(arc) = f (arc) ∗2 g(arc), is
5
an (X; ∗1 ) coloring of X if operations ∗1 , and ∗2 are entropic one with
respect to the other, that is: (a ∗1 b) ∗2 (c ∗1 d) = (a ∗2 c) ∗1 (b ∗2 d).
Proof. For every crossing with initial under-arc a and over-arc b we
need
(f (a) ∗2 g(a)) ∗1 (f (b) ∗2 g(b)) = (f (a) ∗1 f (b)) ∗2 (g(a) ∗1 g(b))
which is exactly the entropic condition in Figure 1.3 (compare Subsection 8.2).
f(b), g(b)
f(a) * 1 f(b), g(a)* 1 g(b)
b
a
f(a), g(a)
f(b) * 2 g(b)
colorings by f and g
f(a) * 2 g(a) * 1 (f(b) *2 g(b)
= f(a) * 1 f(b)
b
* 2 g(a) *1 g(b)
a
f(a) * 2 g(a)
Figure 1.3; Entropy condition for composition of f and g: f ∗2 g
Notice that if g is a trivial coloring, say g(arc) = x for any arc then
any crossing forces x ∗1 x = x and the entropic equation reduces to
(f (a) ∗2 x) ∗1 (f (b) ∗2 x) = (f (a) ∗1 f (b)) ∗2 (x ∗1 x) = (f (a) ∗1 f (b)) ∗2 x
(right distributivity). For use of entropic magmas in Knot Theory, see
[N-P-4, Prz-1, Prz-2, Prz-8, P-T-1, P-T-2, Si]; compare also Proposition
2.6.
We introduce now a monoid of binary operations and show that
distributivity can be studied in the context of this monoid. Then we
compare homology for associative structures (semigroups) with that
Knots and distributive homology
6
for distributive structures (shelves). We also compare extensions in
associative and distributive cases.
The paper is planned as a continuation of pioneering essay [Prz-5]
and for completeness we recall parts of the essay.
2. Monoid of binary operations
Let X be a set and ∗ : X × X → X a binary operation. We call
(X; ∗) a magma6. For any b ∈ X the adjoint map ∗b : X → X, is
defined by ∗b (a) = a ∗ b. Let Bin(X) be the set of all binary operations
on X.
Proposition 2.1. Bin(X) is a monoid (i.e. semigroup with identity)
with the composition ∗1 ∗2 given by a ∗1 ∗2 b = (a ∗1 b) ∗2 b, and the
identity ∗0 being the right trivial operation, that is, a ∗0 b = a for any
a, b ∈ X.
If ∗ ∈ Bin(X) is invertible then ∗−1 is usually denoted by ∗¯. One
should remark that ∗0 is distributive with respect to any other operation, that is, (a ∗ b) ∗0 c = a ∗ b = (a ∗0 c) ∗ (b ∗0 c), and (a ∗0 b) ∗ c =
a ∗ c = (a ∗ c) ∗0 (b ∗ c).
Definition 2.2. Let (X; ∗) be a magma, then:
(i) If ∗ is right self-distributive, that is, (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c),
then (X; ∗) is called an RDS (right distributive structure) or a
shelf (the term coined by Alissa Crans and used in knot theory
[Cr]).
(ii) If a shelf (X; ∗) satisfies the idempotent condition, a ∗ a = a for
any a ∈ X, then it is called an RDI structure or right spindle,
or just a spindle (again the term coined by Crans).
(iii) If a shelf (X; ∗) has ∗ invertible in Bin(X) (equivalently ∗b is a
bijection for any b ∈ X), then it is called a rack (the term wrack,
like in “wrack and ruin”, of J.H.Conway from 1959 [C-W], was
modified to rack in [F-R]).
(iv) If a rack (X; ∗) satisfies the idempotent condition, then it is
called a quandle (the term coined in Joyce’s PhD thesis of 1979
[Joy-1, Joy-2]). Axioms of a quandle were motivated by three
Reidemeister moves (idempotent condition by the first move,
invertibility by the second, and right self-distributivity by the
third move).
6The
term magma was used by J-P. Serre [Ser] and Bourbaki [Bou], replacing
the older term groupoid which started to mean a category with every morphism
invertible.
7
(v) If a quandle (X; ∗) satisfies ∗∗ = ∗0 (i.e. (a ∗ b) ∗ b = a) then
it is called kei or an involutive quandle. The term kei ( ) was
coined in a pioneering paper by M. Takasaki7 in 1942 [Tak]
The main early example of a rack (and a quandle) was a group G
with a ∗ operation given by conjugation, that is, a ∗ b = b−1 ab (Conway
thought of this as a “wrack” of a group). Another example, considered
already in Conway-Wraith correspondence [C-W], is defined for any
group with a ∗ b = ba−1 b and called by Joyce (after Bruck [Bru]) a core
quandle. This example, for a group, H, abelian was already studied
by Takasaki so we call it Takasaki kei (or quandle) and denote by
T (H) (in abelian notation we have a ∗ b = 2b − a), compare [N-P-1].
T (Zn ) is often called a dihedral quandle and denoted by Rn ; it can be
interpreted as composed of reflections of the dihedral group D2n (we
can mention that rack and quandle homology of T (Zn ) for prime n has
been computed in [N-P-2, Cla, Nos]).
More general examples still starting from a group are given in Joyce
paper [Joy-2]:
Example 2.3. Let G be a group and t : G → G a group homomorphism
then we have the following spindle structures on G:
(1) a ∗1 b = t(ab−1 )b,
(2) a ∗2 b = t(b−1 a)b,
(3) If t is invertible both examples give quandles where ¯∗1 and ∗¯2
are given by the formulas:
(i) a¯∗1 b = t−1 (ab−1 )b thus ∗¯1 yielded by the automorphism t is
equal to ∗1 yielded by the automorphism t−1 .
(ii) a¯∗2 b = bt−1 (ab−1 ), e.g. we check that
(a ∗2 b)¯∗2 b = bt−1 ((t(b−1 a)bb−1 ) = a. This example is related to the fundamental group of cyclic (branched) covers of
S 3 along a link. Locally, at every crossing we have relations
C = τ −1 (B −1 A)B and A = Bτ (CB −1 ), as illustrated in Figure
2.1 [Prz-3, Pr-Ro-1, DPT].
(4) If G is an abelian group both examples lead to the same spindle
called Alexander spindle (for t invertible, Alexander quandle).
In abelian notation we get a ∗ b = ta − tb + b = (1 − t)b + ta.
7Mituhisa
Takasaki worked at Harbin Technical University, likely as an assistant
to Kôshichi Toyoda. Both perished when Red Army entered Harbin in August
1945.
Knots and distributive homology
8
This two sided distributive structure was already considered in
1929 by C.Burstin8 and W.Mayer9 [B-M].
(5) If t = −1 we get a ∗ b = 2b − a and this structure, as mentioned
before, was the main example of Kei by Takasaki so we denote
it by T (G).
(6) a ∗3 b = t(ba−1 )b with t2 = t. It is a quandle if and only if
t = Id in which case we get a quandle called the core quandle
of G.
B
C=
−1
(B −1A)B
B
A=B (CB −1 )
Figure 2.1; relations for cyclic covering; see Example 2.3(3)
Definition 2.2 describes properties of an individual magma (X; ∗). It
is also useful to consider subsets or submonoids of Bin(X) satisfying
the related conditions (compare [R-S, Mov, Deh-1, Prz-5].
Definition 2.4. We say that a subset S ⊂ Bin(X) is a distributive
set if all pairs of elements ∗α , ∗β ∈ S are right distributive, that is,
(a ∗α b) ∗β c = (a ∗β c) ∗α (b ∗β c) (we allow ∗α = ∗β ).
(i) The pair (X; S) is called a multi-shelf if S is a distributive set.
If S is additionally a submonoid (resp. subgroup) of Bin(X),
we say that it is a distributive monoid (resp. group).
(ii) If S ⊂ Bin(X) is a distributive set such that each ∗ in S satisfies the idempotent condition, we call (X; S) a multi-spindle.
8Celestyn
Burstin (1888-1938) was born in Tarnopol, where he obtained
“Matura” in 1907, he moved to Vienna where in 1911 he completed university. In
1929 he moved to Minsk where he was a member of the Belarusian National Academy of Sciences, and a Director of the Institute of Mathematics of the Academy.
In December 1937, he was arrested on suspicion of activity as a spy for Poland and
Austria. He died in October 1938, when interrogated in a prison in Minsk (“Minskaja Tjurma”); he was rehabilitated March 2, 1956 [Bur-1, Bur-M, Mal, Mio].
9Walter Mayer (1887–1948) is well known for Mayer-Vietoris sequence and for
being assistant to A.Einstein at Institute for Advanced Study, Princeton [Isa].
9
(iii) We say that (X; S) is a multi-rack if S is a distributive set, and
all elements of S are invertible.
(iv) We say that (X; S) is a multi-quandle if S is a distributive
set, and elements of S are invertible and satisfy the idempotent
condition.
(v) We say that (X; S) is a multi-kei if it is a multi-quandle with
∗∗ = ∗0 for any ∗ ∈ S. Notice that if ∗21 = ∗0 and ∗22 =
∗0 then (∗1 ∗2 )2 = ∗0 ; more generally if ∗n1 = ∗0 and ∗n2 =
∗0 then (∗1 ∗2 )n = ∗0 . This follows from the fact that elements of a multi-quandle commute pairwise (this was observed
by M.Jablonowski [Prz-5]).
Proposition 2.5. [Prz-5]
(i) If S is a distributive set and ∗ ∈ S is invertible, then S ∪ {¯∗}
is also a distributive set.
(ii) If S is a distributive set and M(S) is the monoid generated by
S then M(S) is a distributive monoid.
(iii) If S is a distributive set of invertible operations and G(S) is the
group generated by S, then G(S) is a distributive group.
We show, after G.Mezera [Mez], the fact that any group can be
embedded in Bin(X) for some X, in particular the regular embedding
of G in Bin(G) is given by g → ∗g with a ∗g b = ab−1 gb (compare [Lar]
and Example X.3.15 of [Deh-2] where the operation a ∗g b = ab−1 gb
is called a half-conjugacy). The expression ab−1 gb was also discussed
with respect to free rack by Fenn and Rourke (compare Remark 8.2).
Proposition 2.5 has its analogue for entropic magmas (that is magmas for which (a ∗ b) ∗ (c ∗ d) = (a ∗ c) ∗ (b ∗ d)). More precisely, we say
that a subset S ∈ Bin(X) is an entropic set if for any ∗α , ∗β ∈ S we
have the entropic condition:
(a ∗α b) ∗β (c ∗α d) = (a ∗β c) ∗α (b ∗β d). Then we have:
Proposition 2.6. [N-P-4]
(i) If S is an entropic set and ∗ ∈ S is invertible, then S ∪ {¯∗} is
also an entropic set.
(ii) If S is an entropic set and M(S) is the monoid generated by S
then M(S) is an entropic monoid.
(iii) If S is an entropic set of invertible operations and G(S) is the
group generated by S, then G(S) is an entropic group.
In the next section we consider homology theory of various magmas,
it is useful here to define, for any magma (X; ∗) a supporting structure
which we call an X-set (it is an old concept for (semi)groups and for
quandles it was first considered by S.Kamada).
Knots and distributive homology
10
Definition 2.7. Let (X; ∗) a magma and E a set. We say that E
is an X-set (or right X-set) if there is a function (right action) ∗E :
E × X → E. In general we do not put any conditions on ∗E but if our
magma satisfies some conditions (e.g. associativity or distributivity)
then ∗E should satisfy some related conditions. In particular, we will
look for a magma structure on X ⊔ E having similar structure.
In the following few sections we discuss various homology theories
for magmas (e.g. associative or distributive). In broad approach we
follow [Prz-5] but we stress the use of X-sets in our definitions.
3. Homology of magmas
We survey in this section various homology theories, starting from
homology of abstract simplicial complexes; then we extract (old and
new) properties to define a presimplicial module and a (weak) simplicial module. Further we give two examples of homology for associative
structures (semigroups), and, an important in knot theory, example of
homology for right self-distributive structures (RDS or shelves). Later
we go back to very general notion of homology of a small category
with coefficient in a functor to R-Mod, and recall the notion of a geometric realization in the case of a (pre)simplicial set, and (pre)cubic
set. Reader interested only in distributive homology can go directly to
Section 6.
3.1. Homology of abstract simplicial complexes. Our goal is to
introduce homology of distributive magmas but to keep a historical
perspective we start with the standard (oriented and ordered) homology of abstract simplicial complexes as they provide the framework for
all homology we consider.
Definition 3.1. The abstract simplicial complex K = (V, P ) is a pair
of sets where V = V (K) is called a set of vertices and P (K) = P ⊂ 2V ,
called the set of simplexes of K and it satisfies: elements of P are
finite subsets of V , include all one-element subsets, and if s′ ⊂ s ∈ P
then also s′ ∈ P (that is a subsimplex of a simplex is a simplex).10 A
simplex of n + 1 vertices is called n-dimensional simplex, or succinctly,
n-simplex ( we write s = {vi0 , vi1 , ..., vin }). We define dim(K) as the
maximal dimension of a simplex in K (may be ∞ if there is no bound).
We consider, additionally, the category of abstract simplicial complexes with a class of objects composed of abstract simplicial complexes.
10
Usually we do not allow ∅ as a simplex, but in some situations it is convenient
to allow also an empty simplex, say of dimension −1; it will lead naturally to
augmented chain complexes.
11
Mor(K1 , K2 ) is the set of maps from V (K1 ) to V (K2 ) which send a
simplex to a simplex (that is if f ∈ Mor(K1 , K2 ), s ∈ P (K1 ) then
f (s) ∈ P (K2 )).
We recall here three classical (equivalent) definitions of a homology
of an abstract simplicial complex: ordered, normalized ordered, and
oriented.
Definition 3.2. Recall that a chain complex C is a sequence of modules
over a fixed ring k (here always commutative with identity),
∂n+2
∂n+1
∂
∂n−1
n
C : ... −→ Cn+1 −→ Cn −→
Cn−1 −→ ...
such that ∂n ∂n+1 = 0 (succinctly ∂ 2 = 0). Thus we have im∂n+1 ⊂
ker∂n , and we define homology Hn (C) = ker∂n /im ∂n+1 .
Now for an abstract simplicial complex K = (V, P ) one defines:
(I) (Ordered homology)
Consider a chain complex C ord with k-modules Cnord = CnOrd(C)
a submodule of kV n+1 generated by all sequences (x0 , x1 , ..., xn ),
allowing repetitions, such that the set {x0 , x1 , ..., xn } is a simplex in P (possibly of dimension smaller from n). The boundary
operation is given on the basis by:
n
n
X
X
i
∂(x0 , x1 , ..., xn ) =
(−1) di =
(−1)i (x0 , ..., xi−1 , xi+1 , ..., xn ).
i=0
i=0
The ordered homology of K is defined
Hnord (K, k) = ker∂n /im ∂n+1 .
If k = Z we write Hnord (K).
ord
Notices, that the maps di : Cnord → Cn−1
, (0 ≤ i ≤ n), di (x0 , x1 , ..., xn ) =
(x0 , ..., xi−1 , xi+1 , ..., xn ), called the face maps, satisfy:
(1) di dj = dj−1di for any i < j.
The system (Cn , di ) satisfying the above equality is called a presimplicial module11 and if we limit ourselves to (V n+1 , di ) it is
called a presimplicial set (compare Definition 3.3). The important basic observation isP
that if (Cn , di ) is a presimplicial module
then (Cn , ∂n ), for ∂n = ni=0 (−1)i di , is a chain complex.
Motivation for the boundary operation:
it is coming from the geometrical realization of an abstract simplicial complex as illustrated below (the general setting of geometric realization of a simplicial set is discussed in Section 13):
11The
concept was introduced in 1950 by Eilenberg and Zilber under the name
semi-simplicial complex, [E-Z].
12
Knots and distributive homology
x2
x2
)=
∂(x0 , x1 , x2 ) = ∂(
x0
x1 x0
=
x1
(x1 , x2 ) − (x0 , x2 ) + (x0 , x1 ).
(II) (Normalized ordered homology).
The ordered chain complex allows degenerate simplexes (when
vertices repeat, in particular neighboring vertices repeat). We
define i-degeneracy maps si : Cn → Cn+1 (0 ≤ i ≤ n) by
si (x0 , ..., xn ) = (x0 , ..., xi−1 , xi , xi , xi+1 , ..., xn ). We can check
here easily that:
(2) si sj = sj+1 si , 0 ≤ i ≤ j ≤ n,
sj−1 di if i < j
(3) di sj =
sj di−1 if i > j + 1
(4) di si = di+1 si = IdCn .
(Cn , di , si ) satisfying properties (1)-(4) is called a simplicial module. The notion was introduced by Eilenberg and Zilber in 1950
under the name of complete semi-simplicial complex [E-Z]. It
is convenient to rephrase the definition so it can be used to any
category:
Definition 3.3. Consider a category C, the sequence of objects
Xn , n ≥ 0 and for any n morphisms di , si , 0 ≤ i ≤ n, di ∈
Mor(Xn , Xn−1 ), and si ∈ Mor(Xn , Xn+1 ). We call (Xn ; di , si )
a simplicial category (e.g. simplicial set, simplicial module, or
simplicial space) if the following four conditions hold.
(1) di dj = dj−1 di for i < j,
(2) si sj = sj+1si for i ≤ j,
(3)
si sj = sj+1 si , 0 ≤ i ≤ j ≤ n,
sj−1 di if i < j
d i sj =
sj di−1 if i > j + 1
(4) di si = di+1 si = IdXn .
Eilenberg and Mac Lane proved in 1947 that the degenerate
part of a presimplicial module is an acyclic chain complex (it has
trivial homology) [E-M-1] (the prove was more specific but the
method worked for all simplicial modules defined only 3 years
13
later). We devote Subsection 3.2 to the proof, after [Lod-1], paying attention to which axioms of a simplicial module are used.
In particular, axiom (4) cannot be ignored as the degenerate
chain complex of quandles, which satisfies the property (4) only
partially has often nontrivial homology.
Now back to normalized ordered homology:
Consider submodules CnD (K (named degenerated modules) and
defined by
CnD = span(s0 (Cn−1 ), s1 (Cn−1 ), ..., sn−1(Cn−1 )).
One check that (CnD , ∂n ) is a subchain complex of Cnord (K). Details are given in Subsection 3.2, were it is also proved that this
chain complex is acyclic. We have also quotient chain complex, called normalized ordered chain complex with CnN (K) =
Cn (K)/CnD (K). As homology groups of CnD (K) are trivial we
have isomorphism:
Hnord (K, k) → HnN (K, k).
(III) (Oriented homology). We can consider smaller chain complex
giving the same homology of K by taking the quotient of C N (K)
and considering only “oriented simplexes”. Formally, let C̄n (K)
be a submodule of C N (K) generated by “transposition symmetrizers” (x0 , ...xi−1 , xi , xi+1 , xi+2 , ..., xn )+(x0 , ...xi−1 , xi+1 , xi , xi+2 , ..., xn ).
One checks directly that (C̄n (K), ∂n ) is a subchain complex of
C N (K). The oriented chain complex is the quotient: C ori (K) =
C N (K)/C̄n ((K). It require some effort to prove that the quotient map f : C N (K) → C ori (K) is a chain equivalence and
thus f∗ : H N (K) → H ori (K) is is an isomorphism of homology modules. From this we conclude that all three definitions,
ordered, normalized ordered, and oriented of homology of an
abstract simplicial complex give the same result.
To have more concrete view of oriented chain complex and
oriented homology we order vertices V of K and interpret the
chain group Cnori (C) as a subgroup of ZV n+1 freely generated by
n-dimensional simplexes, (x0 , x1 , ..., xn ) (we assume that x0 <
x1 < ... < xn in our ordering). In essence, with ordering, we are
able to choose representatives of equivalence classes in Cnori (K)
and the boundary operation
∂(x0 , x1 , ..., xn ) =
n
X
i=0
(−1)i di where
Knots and distributive homology
14
di (x0 , x1 , ..., xn ) = (x0 , ..., xi−1 , xi+1 , ..., xn ),
preserves our choice so with given ordering we have a split chain
map g : Cnori (K) → CnN (K). The quotient map, with our ordering can be written as fn (x0 , ..., xn ) = (−1)|π| (x′0 , ..., x′n ) where
π ∈ Sn+1 is the permutation such that x′i = xπ(i) and x′i < x′i+1
(if xi = xj for some i 6= j then we put f (x0 , ..., xn ) = 0). Immediately, we have fn gn = IdCnori(K) . The proof that gf is chain
homotopic to identity on CnN (K) (and so f is chain equivalence)
requires more effort.12
3.2. Degenerate subcomplex. Consider a presimplicial module (Cn , di )
with degenerate maps si . We define degenerate modules
CnD = span(s0 (Cn−1 ), s1 (Cn−1 ), ..., sn−1(Cn−1 )).
We check which conditions are needed so that (CnD , di ) is a sub-presimplicial
module of (Cn , di ). We have:
∂n sp =
n+1
X
i=0
p−1
X
i=0
i
(−1) di sp =
p−1
X
i
p
(−1) di sp +(−1) (dp sp −dp−1 sp +
i=0
(−1)i sp−1 di + (−1)p (dp sp − dp+1 sp +
n+1
X
(3)
(−1)i di sp =
i=p+2
n+1
X
(4′ )
(−1)i sp di−1 =
i=p+2
12The
standard Eilenberg-Mac Lane proof uses acyclic modules method [E-M-2,
Spa], however in our case one can give shorter proof (the idea is still that
of Eilenberg-Mac Lane): consider the chain map f : C N → C ori given by
fn (x0 , ..., xn ) = (−1)|π| (x′0 , ..., x′n ) where π ∈ Sn+1 is the permutation such that
x′i = xπ(i) and x′i < x′i+1 (if xi = xj for some i 6= j then we put f (x0 , ..., xn ) = 0).
The map g : C ori → C N is defined to be embedding; therefore f g = IdC . We show
that gf induces identity on homology of C N . We construct a chain homotopy between gf and the identity inductively, starting from F0 = 0. The main ingredient
of the proof is the fact that for a simplex s = (x0 , ..., xn ) the subchain complex
s̄ = (s, 2s ) of C N is acyclic (HnN (s̄) = 0 for n > 0 and H0N (s̄) = k).
Step n: assume that Fn−1 , ..., F0 are already constructed and we construct a map
N
Fn : CnN → Cn+1
such that ∂n+1 Fn = −Fn−1 ∂n + Id − gf .
We compute: that
∂n (−Fn−1 ∂n + Id − gf ) = −(∂n Fn−1 )∂n + ∂n − ∂n (gf ) =
Fn−2 ∂n−1 ∂n − ∂n + (gf )∂n ∂n − ∂n (gf ) = 0.
Because chain complex CnN (s̄) is exact at place n and −Fn−1 ∂n + Id − gf is in the
kernel of this chain complex, it is also in the image, say, ∂n+1 cn+1 = −Fn−1 ∂n +
Id − gf . Then we declare Fn (s) = cn+1 . In fact here cn+1 can be obtained by
putting any, fixed, vertex of s in front of (−Fn−1 ∂n + Id − gf ). Such constructed
Fn satisfies ∂n+1 Fn + Fn−1 ∂n = Id − gf . Our map is well defined as we constructed
it on the basis of CnN .
15
p−1
n+1
X
X
i
D
(−1) sp−1 di
(−1)i sp di−1 ∈ Cn−1
,
i=0
i=p+2
where the (4’) is the condition:
(4′ ) dp sp = dp+1sp for any p ≤ n.
If (Cn , di , si ) satisfies conditions (1),(2),(3), and (4’) it is called a weak
simplicial module [Prz-5]. As condition (2) was not use in calculation
it is useful also to consider (Cn , di , si ) satisfying the condition (1),(3),
and (4’), we call this a weak-pseudo-simplicial module.
We strengthen the above calculation by considering the sequence of
modules Fni = span(s0 (Cn−1 ), s1 (Cn−1 ), ..., si (Cn−1 )) and the filtration:
0 ⊂ Fn0 ⊂ Fn1 ⊂ ... ⊂ Fnn−1 = CnD .
Our calculation gives ∂n (sp (Cn−1 )) ⊂ span(sp−1 (Cn−1 ), sp (Cn−1)) and
consequently:
Corollary 3.4. If (Cn , di , si ) is a weak-pseudo-simplicial module then
∂n is filtration preserving, that is ∂n (Fnp ) ⊂ Fnp−1 .
We will prove now the Eilenberg-Mac Lane theorem that the degenerate complex (CnD , ∂n ) is acyclic, watching on the way which axioms
are used. For a filtration (Fnp ) the associated graded module is defined
to be {Grnp = Fnp /Fnp−1}. We prove first that the chain complex {Grnp }
is acyclic for any p.
Lemma 3.5. Let (Cn , di , si ) satisfies the conditions (1), (2”), (3), and
(4) where
(2”) sp−1sp−1 = sp sp−1 for every 0 < p ≤ n.
We call such (Cn , di , si ) a co-almost-simplicial module. Then the chain
complex {Grnp = Fnp /Fnp−1 } is acyclic, in particular Hn ({Grnp }) = 0,
and homology of Fnp and Fnp−1 are isomorphic.
Proof. The classical idea of Eilenberg and Mac Lane is to use the degenerate map sp as a chain homotopy (we follow [Lod-1]):
It suffices to show that (∂sp + sp ∂)sp = (−1)p sp modulo sp−1 Cn−1, so
the map sp is a chain homotopy between (−1)p Id and the zero map on
Grnp . In the calculation we stress which axioms are used:
(∂sp + sp ∂)sp = (
n+1
X
i=0
p−1
X
i=0
i
p
i
(−1) di sp + sp
n
X
(−1)i di )sp =
i=0
(−1) di sp sp + (−1) (dp sp − dp+1sp )sp +
n+1
X
(−1)i di sp sp +
i=p+2
Knots and distributive homology
16
p−1
X
i
p
(−1) sp di sp + (−1) sp (dp sp − dp+1 sp ) +
i=0
p−1
X
n
X
(3)
(−1)i sp di sp =
i=p+2
(−1)i sp−1sp−1 di + (−1)p (dp sp − dp+1sp )sp +
i=0
n+1
X
(−1)i sp di−1 sp +
i=p+2
p−1
X
i
p
(−1) sp sp−1di + (−1) sp (dp sp − dp+1 sp ) +
i=0
p−1
X
n
X
(−1)i sp di sp =
i=p+2
(−1)i sp−1 sp−1 di + (−1)p (dp sp − dp+1 sp )sp + (−1)p sp dp+1sp +
i=0
p−1
X
(−1)i sp sp−1 di + (−1)p sp (dp sp − dp+1sp )
dp sp =dp+1 sp
=
i=0
p−1
p−1
X
X
(2)
i
(−1)i sp sp−1 di + (−1)p sp dp+1 sp =
(−1) sp−1 sp−1di +
i=0
i=0
2
p−1
X
(−1)i sp−1sp−1 di + (−1)p sp dp+1sp
mod 2sp−1 Mn−1
=
i=0
(−1)p sp dp+1sp
dp+1 sp =Id
=
(−1)p sp
Now consider the short exact sequence 0 → F p−1 → F p → F p /F p−1 →
0 and the corresponding long exact sequence of homology:
... → Hn+1 (F p /F p−1 ) → Hn (F p−1 ) → Hn (F p ) → Hn (F p /F p−1) → ...
Thus because homology of F p /F p−1 is trivial we obtain isomorphism
Hn (F p−1 ) → Hn (F p ). In conclusion, by induction on p we get the
Eilenberg-Mac Lane result: HnD (C) = 0.
From our proof follows that working modulo 2sp−1Mn−1 , e.g. modulo 2, gives directly Hn (F p ) = 0. Also from axiom (2) we took only
sp sp−1 = sp−1 sp−1 that is axiom (2”).
The above consideration do not work for a distributive case (the
axiom (4) usually does not hold as explained in Section 6 (see [CPP,
Pr-Pu-1, P-S]. We proved however that the degenerate part of quandle
homology can be obtained from the normalized one via Künneth type
formula, see [Pr-Pu-2]).
17
3.3. Bicomplex for a degenerate subcomplex CnD . One more important observation follows from our calculations. If (Cn , di , si ) is a
weak simplicial module13 (i.e. conditions (1)-(3),(4’) hold then the formula
p−1
n
X
X
i
(−1) sp−1di +
(−1)i sp di−1
∂n sp =
i=0
i=p+2
0
allows us to define a bicomplex with entries Ep,q
= Grn,p = Fnp /Fnp−1,
h
n = p + q, with horizontal and vertical boundary operation: ∂p,q
=
Pp−1
P
n
h
i
v
i
h
v
v
i=0 (−1) and ∂p,q =
i=p+2 (−1) sp di−1 with ∂p,q−1 ∂p,q = −∂p−1,q ∂p,q :
0
0
Ep,q
→ Ep−1,q−1
; see Figure 3.1.
↓ ∂v
dh
↓ ∂v
∂h
↓ ∂v
∂h
∂h
0
0
0
. . . ← Ep−1,q+1
← Ep,q+1
← Ep+1,q+1
← ...
v
v
↓∂
↓∂
↓ ∂v
∂h
... ←
∂h
0
Ep−1,q
↓ ∂v
∂h
←
∂h
0
Ep,q
↓ ∂v
∂h
←
0
Ep+1,q
↓ ∂v
∂h
∂h
← ...
∂h
0
0
0
. . . ← Ep−1,q−1
← Ep,q−1
← Ep+1,q−1
← ...
↓ ∂v
↓ ∂v
↓ ∂v
0
Figure 3.1; Bicomplex (Ep,q
, ∂v , ∂h)
0
The bicomplex (Ep,q
, ∂ v , ∂ h ) yields a spectral sequence, in fact two
0 →E 0
ker(∂ v :Epq
p,q−1 )
0
v
0 ),
im(∂ :Ep,q+1 →Epq
h
0
0
ker(∂ :Epq →Ep−1,q )
0
0 )
im(∂ h :Ep+1,q
→Epq
1
spectral sequences: starting from columns, that is c Epq
=
1
and the spectral sequence starting from rows: r Epq
=
which can be used to analyze homology of Gr(C D ) and (Cn ), see
[Pr-Pu-2] for an application in the distributive case.
3.4. Homology with coefficients in a k−Mod functor. Each individual abstract simplicial complex K = (V, P) is a small category14
with simplexes as objects and inclusions of simplexes, s ⊂ s′ , as morphisms. As usually K op will denote the opposite category so restrictions, s ⊃ s′ , are morphisms, more precisely MorK op (s, s′ ) is empty if s
does not contain s′ and otherwise MorK op (s, s′ ) has one morphism denoted by (s ⊃ s′ ). Now for any (covariant) functor F : K op → k−Mod,
where k-Mod is a category of modules over a commutative ring k, we
can define oriented homology Hnori (K, F ) of an abstract simplicial complex K with coefficients in F , as follows:
13In
fact a pseudo weak simplicial module suffices, i.e. conditions (1),(3),(4’).
is called small if objects form a set.
14Category
Knots and distributive homology
18
Definition 3.6. Let K = (V, P ) be an abstract simplicial complex with
ordered vertices15 and F : K op → k−Mod a functor. We define the
presimplicial module (Cnori (K, F ), di) as follows:
M
Cnori (K, F ) =
F (s)
dim(s)=n
ori
(K, F ) is defined by
the face map di : Cnori (K, F ) → Cn−1
di = F (s ⊃ (s − xi )) where s = (x0 , ..., xn ) and xi < xi+1
P
as usually for presimplicial modules ∂n = ni=1 (−1)i di and (Cnori (K, F ), ∂i )
is a chain complex whose homology is denoted by Hnori (K, F ).
The above definition can be thought of as a twisted version of an
oriented homology of abstract simplicial complexes. Similarly we can
define ordered homology of (K, F ) but oriented and ordered homology
with coefficient in a functor are not necessarily isomorphic.
Definition 3.6 is related to more general Definition 3.7 on homology
of a small category with a functor coefficient, usually thought to be
first given by [Wat], who in turn refers to the earlier paper [Dehe] in
the case of the category of posets.
Definition 3.7. Let P be as small category (i.e. objects, P = Ob(P)
form a set), and let F : P → k-Mod be a functor from P to the category
of modules over a commutative ring k. We call the sequence of objects
fn−1
f0
f1
and functors, x0 → x1 → . . . → xn an n-chain (more formally n-chain
in the nerve of the category). We define the chain complex C∗ (P, F )
as follows:
M
Cn =
F (x0 )
f
f
fn−1
0
1
x0 →x
1 →... → xn
where the sum is taken over all n-chains.
The boundary operation ∂nP: Cn (P, F ) → Cn+1 (P, F ) is an alternative sum of face maps, ∂n = ni=0 (−1)i di , where di are given by:
f0
f1
fn−1
f0
f1
fn−1
d0 (λ; x0 → x1 → . . . → xn ) = (F (x0 → x1 )(λ); x1 → . . . → xn ),
f0
f1
fn−1
f0
f1
fi fi−1
fn−1
di (λ; x0 → x1 → . . . → xn ) = (λ; x0 → x1 → . . . → xi−1 → xi+1 → . . . → xn )
15It
suffices to have V partially ordered as long as for any simplex s = (x0 , ..., xn )
the partial order on V restricts to linear order on vertices of s. Even better we do
not need a partial order, it suffices that vertices of every simplex are ordered in
such a way that if s1 ⊂ s2 then the ordering of vertices of s1 is a restriction of the
ordering of vertices of s2 .
19
for 0 < i < n, and
f1
f0
fn−1
f0
f1
fn−2
dn (λ; x0 → x1 → . . . → xn ) = (λ; x0 → x1 → . . . → xn−1 ).
We denote by Hn (P, F ) the homology yielded by the above chain complex, and call this the homology of a small category P with coefficients
in a functor F .
Similarly, if F ′ : P → k−Mod is a contravariant functor we may define
a homology Hn (P; F ′ ), starting from
M
Cn (P; F ′ ) =
F ′ (xn ).
f
f
fn−1
0
1
x0 →x
1 →... → xn
f0
fn−1
f1
In particular, dn (x0 → x1 → . . . → xn ; λ) =
f0
f1
fn−2
fn−1
(x0 → x1 → . . . → xn−1 ); F ′ (xn−1 → xn )(λ)), where λ ∈ F ′ (xn ).
One can also consider both functors, F and F ′ in the definition starting
from
M
Cn (P; F , F ′) =
F ′(xn ) ⊗ F (x0 );
f
f
fn−1
0
1
x0 →x
1 →... → xn
compare Definition 4.6. We can also start from from a bifunctor D :
P op × P → k − Mod and mimic the definition of the Hochschild homology (Section 5) [Lod-1].
Remark 3.8. Any subcategory P ′ of P has its chain complex, and
homology (we use the functor F ′ = F /P ′ , that is, the restriction of
F to P ′ ). C∗ (P ′ , F ′) is a subchain complex of C∗ (P, F ) so we can
consider the short exact sequence of chain complexes:
0 → Cn (P, F ) → Cn (P ′ , F ′ ) → Cn (P, F )/Cn (P ′ , F ′ ) → 0
and yielded by it the long exact sequence of homology.
The pair (Cn , di ) form a presimplicial module by associativity of
morphisms of a category and properties of a functor. More generally
we have:
Proposition 3.9. Let si : Cn → Cn+1 be a map inserting identity morphism on the ith place in the nth chain of of the nerve of the category,
that is
f0
fn−1
f0
Idx
fn−1
si ((λ; x0 → . . . → xn ) = (λ; x0 → xi →i xi . . . → xn ).
Then (Cn , di , si ) is a simplicial module.
The classical example is the homology of a simplicial complex with
constant coefficients, that is F (s) = k and F (f ) = Idk ; in that case we
write Hn (K, F ) = Hn (K, k) or just Hn (K) if k = Z. Related to this
Knots and distributive homology
20
example is homology of posets: if P is a small category and for any
objects x and y, Mor(x, y) has at most one element and additionally if
Mor(x, y) 6= ∅ and Mor(y, x) 6= ∅ then x = y, then P is a poset with
x ≤ y iff Mor(x, y) 6= ∅.
Another classical example concerns homology of groups, where the
category has one object and G morphisms (interpreted as multiplication by elements of G [Bro], that is the morphism g : G → G is given
by g(h) = hg); compare Section 4.
More recent example is motivated by Khovanov homology so we call
a related functor F(D,A,M ), a Khovanov functor. The functor depends
on a choice of an k-Frobenius algebra A, A-Frobenius bimodule M and
a link diagram (possibly virtual link, or a link diagram on a surface, L
(equivalently we can work with graphs on a surface). Here for simplicity
we assume that M = A is an abelian Frobenius k-algebra16 and D is a
classical link diagram.
Definition 3.10. For a link diagram D, let V be the set of its crossings
(in some order), and P = 2V . Thus K = (V, P ) is a simplex (we allow
also the empty, −1-dimensional simplex). Let A be a Frobenius algebra
with a multiplication
µ and a co-multiplication ∆ (e.g. A = Z[x]/(xm ),
P
∆(1) = i+j=m−1 xi ⊗ xj ). We define a functor FD,A : K → k-Mod
as follows. For any s ∈ P we identify s with a Kauffman state, where
s(v) = 1 (i.e.
) iff v ∈ s. We denote by Ds the collection of
circles obtained from D by smoothing it according to s, and by |Ds | the
number of circles in Ds . Then we define F (s) = A⊗|Ds | . To define
F (s ⊃ (s − vi )) we first decorate circles of Ds by algebra A, (that is
each circle by one copy of A); then we have two cases:
(µ) |Ds−vi | = |Ds | − 1, thus two circles are glued together when we
switch the state at vi . In this case we multiply the element associated
with glued circles (we use the fact that A is commutative).
(∆) |Ds−vi | = |Ds | + 1, thus a circle of Ds is split so we apply to the
element of A associated to this circle a co-multiplication (we use the
fact that ∆ is co-commutative). F is a functor as A is a commutative
Frobenius algebra.
This approach to Khovanov homology was first sketched in [Prz-4],
where Khovanov homology was connected to Hochschild homology.
16A
is a k module with associative and commutative multiplication, µ, with
co-associative and co-commutative co-multiplication, ∆, satisfying the Frobenius
condition, that is ∆µ = (µ × Id)(Id × ∆); graphically:
. There is no need for
unit and counit.
21
It is a classical result that homology of a baricentric subdivision
of an abstract simplicial complex is isomorphic to the homology of
the complex. This was an ingredient of original proofs of topological
invariance of homology. The generalization of the result holds also for
a homology of a simplicial complex K with a coefficient in a functor
and the homology of K treated as a small category. I was informed by
S. Betley and Jolanta Slomińska about at least three proof of the fact,
compare [Slo]. We are writing, with my student Jing Wang detailed
survey with the proof following closely the classical proof with constant
coefficients (in essence it is another case of acyclic model theorem of
Eilenberg and Zilber [E-Z]).
In the next few sections we discuss homology related to various magmas (e.g. associative and right distributive) and look for the common
traits, for example presimplicial or simplicial module structure, geometric realization etc.
4. Group homology of a semigroup
In the homology of abstract simplicial complexes, the set of vertices,
X, has no algebraic structure or, as we will see later, we can treat X as
a magma with the trivial operation ∗0 , x ∗0 y = x. We will now discuss
homology of magma (X, ∗) equipped with some specific structure, e.g.
associativity, Jacobi identity, or distributivity.
According to [Bro]: The cohomology theory of groups arose from both
topological and algebraic sources. The starting point for the topological
aspect of the theory was the work of Hurewicz ([Hur], 1936 on “aspherical spaces”. About a year earlier, Hurewicz had introduced the
higher homotopy groups πn X of a space X (n ≥ 2). He now singled
out for study those path-connected spaces X whose higher homotopy
groups are all trivial, but whose fundamental group π = π1 X need not
be trivial. Such spaces are called aspherical. Hurewicz proved, among
other things, that the homotopy type of an aspherical pace X is completely determined by its fundamental group π. ... Hopf ([Hop], 1942)...
expressed H2 π in purely algebraic terms...
Let (X, ∗) be a semigroup that is a set with associative binary operation. We associate with (X, ∗) a presimplicial set, presimplicial module,
chain complex, group homology and geometric realization as follows:
Definition 4.1.
(i) Let Xn = X n and di : Xn → Xn−1 for 0 ≤ i ≤
n is given by:
d0 (x1 , x2 , ..., xn ) = (x2 , , ..., xn ),
di (x1 , ..., xn ) = (x1 , ..., xi−1 , xi ∗ xi+1 , xi+2 , ..., xn ) for 0 < i < n,
Knots and distributive homology
22
d0 (x1 , ..., xn−1 , xn ) = (x1 , , ..., xn−1 ).
Then (Xn , di ) is a presimplicial set.
(ii) If we choose a commutative ring k and consider Cn = kX n and
di : Cn → Cn−1 the unique extension of the map di from (i)
then (CnP
, di ) is a presimplicial module.
(iii) If ∂n = ni=0 (−1)i di , then (Cn , ∂n ) is a chain complex; its homology are called group homology of a semigroup X and denoted
by Hn (X; k) or just Hn (X) if k = Z.
(iv) A presimplicial set has a standard geometric realization, BX (as
a CW-complex17). Thus the semigroup homology has a natural
interpretation as a homology of a CW-complex [Bro].
Definition 4.1 has a classical generalization, when a semigroup (X; ∗)
is augmented by an X-right-semigroup-set E, that is a set with the right
action (also denoted by ∗) of X on E such that (e ∗ a) ∗ b = e ∗ (a ∗ b).
Definition 4.2.
(i) Let Xn = E × X n and di : Xn → Xn−1 for
0 ≤ i ≤ n is given by:
d0 (e, x1 , x2 , ..., xn ) = (e ∗ x1 , x2 , , ..., xn ),
di (e, x1 , ..., xn ) = (e, x1 , ..., xi−1 , xi ∗ xi+1 , xi+2 , ..., xn ) for 0 < i < n,
dn (e, x1 , ..., xn−1 , xn ) = (e, x1 , , ..., xn−1 ).
Then (Xn , di ) is a presimplicial set.
(ii) For Cn = k(E × X n ), (Cn , di ) is a presimplicial module and
(Cn , ∂n ) is a chain complex with homology denoted by Hn (X, E)
and geometric realization B(X, E).
If E has one element then we get the case of Definition 4.1.
Definition 4.2 has further generalization if, in addition to a semigroup
(X; ∗) we have the right X-set E0 and the left X set Ew (here we need
(a ∗ (b ∗ e) = (a ∗ b) ∗ e), compare [Ca-E], Chapter X.
Definition 4.3.
(i) Let Xn = E0 × X n × Ew and di : Xn → Xn−1
for 0 ≤ i ≤ n is given by:
d0 (e0 , x1 , x2 , ..., xn , en+1 ) = (e ∗ x0 , x2 , , ..., xn , en+1 )),
di (e0 , x1 , ..., xn , en+1 ) = (e0 , x1 , ..., xi−1 , xi ∗xi+1 , xi+2 , ..., xn , en+1 ) for 0 < i < n,
dn (e0 , x1 , ..., xn−1 , xn , en+1 ) = (e0 , x1 , ..., xn−1 , xn ∗ en+1 ).
Then (Xn , di ) is a presimplicial set. We call this pre-simplicial
set a “two walls” presimplicial set due to visualization in Figure
4.1.
17BX
can be made into geometric simplicial complex by second baricentric subdivision because BX by the construction is glued from simplexes (such a space is
called a ∆-complex in [Hat]), see Section 13, e.g. Definition 13.1
23
(ii) For Cn = k(E0 × X n × Ew ), (Cn , di ) is a presimplicial module and (Cn , ∂n ) is a chain complex with homology denoted by
Hn (X, E0 , Ew ), and geometric realization B(X, E0 , Ew ).
If Ew has one element then we get the case of Definition 4.2
E0
X
i
i+1
X
E w
Figure 4.1; i’th face map in “two walls” presimplicial set for a semigroup
A version of Definition 4.3 when we assume that E0 = Ew and (e1 ∗
x) ∗ e2 = e1 ∗ (x ∗ e2 ) that is E0 is an X-biset leads to the Hochschild
homology. In particular, for a semigroup (X; ∗) we have:
Definition 4.4. Let (X; ∗) be a semigroup and E an X-biset then:
(i) Let Xn = E × X n and di : Xn → Xn−1 for 0 ≤ i ≤ n is given
by:
d0 (e, x1 , x2 , ..., xn ) = (e ∗ x1 , x2 , , ..., xn ),
di (e, x1 , ..., xn ) = (e, x1 , ..., xi−1 , xi ∗ xi+1 , xi+2 , ..., xn ) for 0 < i < n,
dn (e, x1 , ..., xn−1 , xn ) = (xn ∗ e, x1 , , ..., xn−1 ).
Then (Xn , di ) is a presimplicial set.
(ii) For Cn = k(E × X n ), (Cn , di ) is a presimplicial module and
(Cn , ∂n ) is a chain complex with (Hochschild) homology denoted
by HHn (X, E) and geometric realization BH(X, E).
(iii) If E0 is an X-right-semigroup-set and Ew is an X-right-semigroupset then we can take E = Ew ×E0 and E has a natural structure
of an X-biset. Thus the concepts of a “two-wall” semigroup homology and Hochschild semigroup homology are equivalent.
If (X; ∗) is a monoid (with a unit element 1) then we say that the
set E is X-right-monoid-set if it is X-right-monoid-set and additionally
e ∗ 1 = e for any e ∈ E (that is 1 acts trivially on E from the right.
Similarly we define X-left-monoid-set (e.g. 1 ∗ e = e). For a monoid
the presimplicial sets (modules) described in Definitions 4.2, 4.3, and
4.4 are in fact simplicial sets (modules) with the degeneracy maps si
placing 1 between xi and xi+1 , for example in the case of 4.3:
s0 (e0 , x1 , x2 , ..., xn , en+1 ) = (e0 , 1, e0 , x1 , x2 , ..., xn , en+1 ),
24
Knots and distributive homology
si (e0 , x1 , x2 , ..., xn , en+1 ) = (e0 , x1 , ..., xi , 1, xi+1 , ..., xn , en+1 ) for 0 < i < n,
sn (e0 , x1 , x2 , ..., xn , en+1 ) = (e0 , x1 , x2 , ..., xn , 1, en+1 ).
Example 4.5. We can check that di di+1 = di di (0 < i < n) if and only
if ∗ is associative.
Furthermore, d0 d1 = d0 d0 iff (e0 ∗ x1 ) ∗ x2 = e0 ∗ (x1 ∗ x2 ) that is E0 is
an X-right-semigroup-set.
Similarly dn−1 dn = dn−1dn−1 iff Ew is an X-left-semigroup-set.
Let ∂ (ℓ) be a boundary map obtained from the group homology
boundary operation by dropping the first term from the sum. Analogously, let ∂ (r) be a boundary map obtained from the group homology
boundary operation by dropping the last term from the sum. It is a
classical observation that (Cn , ∂ (ℓ ) and (Cn , ∂ (r) ) are acyclic for a group
(or a monoid). We show this in a slightly more general context of weak
simplicial modules (used later in the distributive case) in Section 6. It
would be of interest to analyze homology of (Cn , ∂ (ℓ ) for a semigroup
without identity. Can it have a torsion?.
Our definition (in the presented form) can be generalized to any kalgebra V not only V = kX Below we give the definition for “two
wall” k-algebra, and in the next section we describe the mainstream
Hochschild homology of k-algebra closely related to group homology.
Definition 4.6. Let A be a k-algebra which acts from the right on a
k-module M0 ((m ∗ x) ∗ y = m ∗ (x ∗ y)) and from the left on a k-module
Mw (x ∗ (y ∗ m) = (x ∗ y) ∗ m), that is M0 is a right A-module and
Mw a left A-module. We define chain groups Cn = M0 ⊗ A⊗n ⊗ Mw
and face maps di (x0 , x1 , ..., xn , xn+1 ) = (x0 , ..., xi ∗ xi+1 , ..., xn+1 ), 0 ≤
i ≤ n, x0 ∈ M0 , xn+1 ∈ Mw , and xi ∈ A for 0 < i P
≤ n. Then
(Cn , di ) is a presimplicial module and (Cn , ∂n ), with ∂n = ni=0 (−1)i di
is a chain complex, whose homology is denoted by Hn (A, M0 , Mw ). If
A is a unitary algebra (with unit 1) then we define degenerate maps
si (x0 , x1 , ..., xn , xn+1 ) = (x0 , x1 , ..., xi , 1, xi+1 , ..., xn , xn+1 ), 0 ≤ i ≤ n,
and one checks directly that (Cn , di , si ) is a simplicial module.
If we glue together M0 and Mw to get 2-sided module (A-bimodule)
M = Mw ⊗ M0 we obtain Hochschild homology Hn (A, M), [Hoch,
Lod-1]; see Section 5.
5. Hochschild homology of a semigroup and an algebra
Hochschild homology was created to have a homology theory of
algebras, as before homology was defined only for (semi)groups, G,
and (semi)group algebras kG (Definition 4.6 is only afterthought with
Hochschild homology in mind). The history of discovering homology for
25
algebra is described in Mac Lane autobiography [Mac]: Given his topological background and enthusiasm, Eilenberg was perhaps the first person to see this clearly. He was in active touch with Gerhard Hochschild,
who was then a student of Chevalley at Princeton. Eilenberg suggested
that there ought to be a cohomology (and a homology) for algebras.
This turned out to be the case, and the complex used to describe the
cohomology of groups (i.e. the bar resolution) was adapted to define
the Hochschild cohomology of algebras.
Nevertheless, we start from Hochschild homology of semigroups as
it leads to a presimplicial set, while the general Hochschild homology
gives a presimplicial module.
Let (X; ∗) be a semigroup and E a two sided X-semigroup-set that
is (e ∗ a) ∗ b = e ∗ (a ∗ b), (a ∗ e) ∗ b = a ∗ (e ∗ b), and (a ∗ b) ∗ e = a ∗ (b ∗ e)
We define a Hochschild presimplicial module {Cn (X, E), di } as follows
[Hoch, Lod-1]: Cn (X) = k(E × X n ) and the Hochschild face map
is given by di : k(E × X n ) → k(E × X n−1 where d0 (e0 , x1 , ...xn ) =
(e0 ∗ x1 , x2 , ..., xn ),
di (e0 , x1 , ...xn ) = (e0 , x1 , ..., xi−1 , xi ∗ xi+1 , ..., xn for 0 < i < n, and
dn (e0 , x1 , ...xn ) = (xn ∗ e0 , x1 , ..., xn−1 ).
∂n : ZX n → ZX n−1 is defined by:
∂(x0 , x1 , ...xn ) =
n−1
X
(−1)i (x0 , ..., xi−1 , xi ∗ xi+1 , xi+2 , ..., xn )+
i=0
(−1)n (xn ∗ x0 , x1 , ...xn−1 )
The resulting homology is called the Hochschild homology of a semigroup (X, ∗) and denoted by HHn (X) (introduced by Hochschild in
1945 [Hoch]). It is useful to define C−1 = Z and define ∂0 (x) = 1
to obtain the augmented Hochschild chain complex and augmented
Hochschild homology.
Again if (X, ∗) is a monoid then dropping the last term gives an
acyclic chain complex.
P
Notice that ∂n = ni=0 (−1)i di , where di (x0 , ..., xn ) = (x0 , ..., xi−1 , xi ∗
xi+1 , xi+2 , ..., xn ), for 0 ≤ i < n and
dn (x0 , ..., xn ) = (xn ∗ x0 , ..., xn−1 ).
Again, (Cn , di ) is a presimplicial module. If (X, ∗) is a monoid, one
can define n + 1 homomorphisms si : Cn → Cn+1 , called degeneracy maps, by si (x0 , ..., xn ) = (x0 , ..., xi , 1, xi+1 , ..., xn ) (similarly, in
the case of group homology of a semigroup, we put, si (x1 , ..., xn ) =
(x1 , ..., xi , 1, xi+1 , ..., xn )). We check that in both cases the following
Knots and distributive homology
26
conditions hold:
(1) di dj = dj−1di f or i < j.
(2) si sj = sj+1 si , 0 ≤ i ≤ j ≤ n,
sj−1di if i < j
(3) di sj =
sj di−1 if i > j + 1
(4) di si = di+1 si = IdCn .
(Cn , di , si ) satisfying conditions (1)-(4) above is called a simplicial
module18 (e.g. Z-module/abelian group). If we replace (4) by a weaker
condition di si = di+1 si we deal with a weak simplicial module, the concept useful in the theory of homology of distributive structures (spindles
or quandles).
As we already mentioned before Hochschild homology (and presimplicial module) can be defined for any algebra A and two-sided Amodule M. We put Cn (A; M) = M ⊗ A⊗n and di (m, x1 , ..., xn ) is given
by:
d0 (m, x1 , ..., xn ) = (mx1 , x2 , ..., xn ).
di (m, x1 , ..., xn ) = (m, x1 , ..., xi−1 , xi xi+1 , xi+2 , ..., xn ) for 0 < i < n, and
dn (m, x1 , ..., xn ) = (xn m, x1 , ..., xn−1 ).
From associativity of A and our action of A on M follows that (Cn , di )
is a presimplicial module. Furthermore, if A is unitary we can define
a simplicial module structure (Cn , di , si ), by putting si (m, x1 , ..., xn ) =
(m, x1 , ..., xi−1 , 1, xi , ..., xn ).
6. Homology of distributive structures
Recall that a shelf (or right distributive system (RDS)) (X; ∗) is a
set X with a right self-distributive binary operation ∗ : X × X → X
(i.e. (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c)).
We work, for simplicity, with chain complexes and homology over Z
but we can replace Z by any commutative ring k in our considerations.
We start from atomic definition, one term distributive homology,
introduced in 2010 just before Knots in Poland III conference [Prz-5].
18The
concept of a simplicial set was introduced by Eilenberg and Zilber who
called it complete semi-simplicial complex; their semi-simplicial complex is now
usually called presimplicial set [E-Z, May].
27
6.1. One-term distributive homology.
Definition 6.1. We define a (one-term) distributive chain complex C (∗)
(∗)
as follows: Cn = ZX n+1 and the boundary operation ∂n : Cn → Cn−1
is given by:
∂n(∗) (x0 , ..., xn ) = (x1 , ..., xn )+
n
X
(−1)i (x0 ∗ xi , ..., xi−1 ∗ xi , xi+1 , ..., xn ).
i=1
The homology of this chain complex is called a one-term distributive
(∗)
homology of (X; ∗) (denoted by Hn (X)).
We directly check that ∂ (∗) ∂ (∗) = 0.
(∗)
We can put C−1 = Z and ∂0 (x) = 1. We have ∂0 ∂1 = 0, so we
obtain an augmented distributive chain complex and an augmented
(∗)
(one-term) distributive homology, H̃n . As in the classical case we get:
(
(∗)
Z ⊕ H̃n (X) n = 0
(∗)
Proposition 6.2. Hn (X) =
(∗)
H̃n (X)
otherwise
(∗)
If (X; ∗) is a rack then the complex (Cn , ∂ (∗) ) is acyclic, but in
the general case of a shelf or spindle homology can be nontrivial with
nontrivial free and torsion parts (joint work with A.Crans, K.Putyra
and A.Sikora [CPP, Pr-Pu-1, P-S]).
If we define di : Cn → Cn−1 , 0 ≤ i ≤ n, by di (x0 , ..., xn ) =
(x0 ∗ xi , ..., xi−1 ∗ xi , xi+1 , ..., xn ), then (Cn , di ) is a presimplicial module and X n+1 , di ) is a presimplicial set. If we define degeneracy maps
si (x0 , ..., xn ) = (x0 , .., xi−1 , xi , xi , xi+1 , ...xn ) then one checks that (Cn , di , si )
is a very weak simplicial module. If we assume idempotency, that
is (X; ∗) is a spindle, then (Cn , di , si ) is a weak simplicial module
and the degenerate part (CnD , ∂n ) is a subchain complex which splits
from Cn , ∂n ) (see [Prz-5]. This split is analogous to the one conjectured in [CJKS] and proved in [L-N] for classical quandle homology. In [N-P-2] we gave very short, easy to visualize and to generalize, proof using the split map CnN → Cn given by (x0 , x1 , ..., xn ) →
(x0 , x1 − x0 , ..., xn − xn−1 ).
We can repeat our definitions if (X; ∗) is a shelf and Y is a shelf-set
(∗ : Y × X → Y with (y ∗ x1 ) ∗ x2 = (y ∗ x2 ) ∗ (x1 ∗ x2 ) see Figure 6.1
for visualization). The presimplicial set (Y × X n+1 , di ) has face maps
di defined by di (y, x0, ..., xn ) = (y ∗ xi , x0 ∗ xi , ..., xi−1 ∗ xi , xi+1 , .., xn ).
The face map di is visualized in Figure 6.2; this visualization will play
an important role when distributive homology will be generalized to
Yang-Baxter homology.
Knots and distributive homology
28
Y
X
X
Y
X
X
Figure 6.1; Graphical interpretation of the axiom for X-shelf-set Y
(y ∗ x1 ) ∗ x2 = (y ∗ x2 ) ∗ (x1 ∗ x2 )
( *)
X
X
X
i
di
Y
X
X
X
i
(∗)
Figure 6.2; Graphical interpretation of the face map di
6.2. Multi-term distributive homology. The first homology theory
related to a self-distributive structure was constructed in early 1990s
by Fenn, Rourke, and Sanderson [FRS-2] and motivated by (higher
dimensional) knot theory19. For a rack (X, ∗), they defined rack homology HnR (X) by taking CnR = ZX n and ∂nR : Cn → Cn−1 is given
(∗)
(∗0 )
by ∂nR = ∂n−1 − ∂n−1
. Our notation has grading shifted by 1, that is,
R
R
Cn (X) = Cn+1
= ZX n+1 . It is routine to check that ∂n−1
∂nR = 0. However, it is an interesting question what properties of ∗0 and ∗ are really
used. With relation to the paper [N-P-3] we noticed that it is distributivity again which makes (C R (X), ∂nR ) a chain complex. More generally
we observed that if ∗1 and ∗2 are right self-distributive and distributive
with respect to each other, then ∂ (a1 ,a2 ) = a1 ∂ (∗1 ) + a2 ∂ (∗2 ) leads to a
chain complex (i.e. ∂ (a1 ,a2 ) ∂ (a1 ,a2 ) = 0). Below I answer a more general
question: for a finite set {∗1 , ..., ∗k } ⊂ Bin(X) and integers a1 , ..., ak ∈
Z, when is (Cn , ∂ (a1 ,...,ak ) ) with ∂ (a1 ,...,ak ) = a1 ∂ (∗1 ) + ... + ak ∂ (∗k ) a chain
(a ,...,a )
complex? When is (Cn , di 1 k ) a presimplicial set? We answer these
19The
recent paper by Roger Fenn, [Fenn] states: ”Unusually in the history of
mathematics, the discovery of the homology and classifying space of a rack can be
precisely dated to 2 April 1990.”
29
questions in Lemma 6.3. In particular, for a distributive set {∗1 , ..., ∗k }
the answer is affirmative.
Lemma 6.3.
(i) If ∗1 and ∗2 are right self-distributive operations, then (Cn , ∂ (a1 ,a2 ) )
is a chain complex if and only if the operations ∗1 and ∗2 satisfy:
6.4.
(a ∗1 b) ∗2 c + (a ∗2 b) ∗1 c = (a ∗2 c) ∗1 (b ∗2 c) + (a ∗1 c) ∗2 (b ∗1 c) in ZX.
We call this condition weak distributivity. If Condition 6.4
does not hold we can take C0 (X) to be the quotient by Equation
6.4: C0 (X) = ZX/6.4 and then we take Cn = C0⊗n+1 .
(ii) We say that a set {∗1 , ..., ∗k } ⊂ Bin(X) is weakly distributive if
each operation is right self-distributive and each pair of operations is weakly distributive (with two main cases: distributivity
(a ∗1 b) ∗2 c = (a ∗2 c) ∗1 (b ∗2 c) and chronological distributivity20
(a ,...,a )
(a ∗1 b) ∗2 c = (a ∗1 c) ∗2 (b ∗1 c)). We have: (Cn , di 1 k ) is a
presimplicial set if and only if the set {∗1 , ..., ∗k } ⊂ Bin(X) is
weakly distributive.
(a ,...,a )
(iii) (Cn , ∂n 1 k ) is a chain complex if and only if the set {∗1 , ..., ∗k } ⊂
Bin(X) is weakly distributive.
We complete this section by showing that for a rack homology of a
R
quandle or spindle, HnR embeds in Hn+1
; we construct monomorphic
21
“homology operation” of degree one . We place it in a more general
context of weak simplicial modules.
Lemma 6.5. Let (Cn , di , si ) be a weak simplicial module then ∂s0 +
s0 ∂ = s0 d0 ; in effect s0 d0 induces a trivial map on homology. In particular:
(i) if the map s0 d0 is the identity then s0 d0 s0 = s0 (as in the case of
a simplicial module) and then the chain complex (s0 (Cn−1 ), ∂n )
is acyclic,
(ii) if d0 = 0 then s0 is a chain map (e.g. this hold for 2-term rack
homology),
(iii) In the case of one term distributive homology, we conclude that
the map replacing x0 by x1 in (x0 , x1 , ..., xn ) is a chain map,
chain homotopic to zero map.
20I
did not see this concept considered in literature, but it seems to be important
in K.Putyra’s work on odd Khovanov homology [Put].
21For a quandle it is a well know fact that rack homology in dimension n is
isomorphic to “early degenerate” homology in dimension n + 1.
Knots and distributive homology
30
Proof. We have ∂s0 + s0 ∂ =
d 0 s0 − d 1 s0 +
n+1
X
i
(−1) di s0 + s0 d0 +
d 0 s0 − d 1 s0 +
i=2
(−1)i s0 di =
i=1
i=2
n+1
X
n
X
n
X
(−1) s0 di−1 + s0 d0 +
(−1)i s0 di = s0 d0 .
i
i=1
If s0 has a left inverse map, say pn : Cn+1 → Cn , pn s0 = IdCn , as is
the case for a weak simplicial module in (multi) spindle case, we can
say more.
Lemma 6.6. Let (Cn , di , si ) be a weak simplicial module pn : Cn+1 →
Cn is a left inverse of s0 and additionally pn di = di−1 pn for i > 0 then
p∂ + ∂p = pd0 ; in effect pd0 induces a trivial map on homology. In
particular:
(i) If we deal with (multi)term distributive homology, p may be
taken to be the map deleting the first coordinate of (x0 , x1 , ..., xn )
(in one term distributive homology p = d0 , so d0 d0 is a chain
map trivial on homology).
(ii) if d0 = 0 then p is a chain map ((e.g. this hold for 2-term rack
homology),
(iii) if d0 = 0 then p induces an epimorphism on homology and s0
induces a monomorphism on homology; in particular s0 induces
monomorphic “homology operation” of degree one (s0 (Cn−1 , ∂n )
is called an early degenerate chain complex).
Proof. We have p∂ + ∂p =
pd0 +
n
X
i=1
(−1)i di−1 p +
n−1
X
(−1)i di p = pd0
i=0
Part (iii) follow from the fact that pn s0 = IdCn and p and s0 are chain
maps.
Thus we proved that rack homology of quandles (or spindles) cannot
R
decrease with n (HnR ⊂ Hn+1
).
We computed with K.Putyra [Pr-Pu-1] various multi-term homology, including that for finite distributive lattices (including Boolean
algebras).
31
7. Bloh-Leibniz-Loday algebra
Lie algebra was probably the first nonassociative structure for which
homology was defined [Ch-E]. The idea of Chevalley and Eilenberg was
to translate homology of a (Lie) group to homology of its Lie algebra22.
We should stress, in particular the role of conjugacy in Lie algebra, as
conjugacy was the motivation for wracks (racks) and quandles.
We discuss here homology theory of Bloh-Loday-Leibniz algebras
introduced by Bloh an Loday [Blo-1, Blo-2, Lod-2], and which can be
informally thought to be a linearization of distributive homology.23 We
follow Loday and Lebed here [Lod-1, Lod-2, Leb-1, Leb-2]. Because
BLL (Bloh-Leibniz-Loday) algebra is a generalization of a Lie algebra
we use a bracket [−, −] for a bilinear map:
Definition 7.1.
(1) Let V be a k-module equipped with a bilinear
map [−, −] : V ×V → V satisfying the relation (Leibniz version
of the Jacobi identity):
[x, [y, z]] = [[x, y], z] − [[x, z], y], for all x, y, z ∈ V .
I see the linearization of distributivity as
(x∗y)∗z = (x∗z)∗(y∗z) =⇒ (x∗y)∗z = (x∗z)∗y+x∗(y∗z) BLL condition.
22The
paper starts from: The present paper lays no claim to deep originality.
Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups maybe reduced to algebraic questions
concerning Lie algebras.
23BLL algebras are often call Leibniz algebras as the version of Jacobi identity
they satisfy can be treated as a Leibniz rule. The history of the discovery is described by Loday as follows [Lod-1]:
“In the definition of the Chevalley-Eilenberg complex of a Lie algebra G the module
of chains is the exterior module. The non-commutative analog
of the exterior modN
ule ΛG is the tensor module T G. If one replaces Λ by
in the classical formula
for the boundary map d of the CE-complex, then one gets a well defined map T G
but the relation d2 is not valid anymore. However I discovered that, if one writes
d so as as to put the commutator [xi , xj ] at the place i when i < j,..., then the
relation d2 = 0 is satisfied in the tensor (i.e. non-commutative) context. So, this
give rise to a new complex T G, d) for the Lie algebra G. The homology groups
of this complex are denoted are denoted HL∗ (G) and called the non-commutative
homology groups of G. In the proof of the relation d2 = 0 in the tensor module case,
I noticed that the only property of the Lie bracket, which is needed, is the Leibniz
relation [x, [y, z]] = [[x, y], z] − [[x, z], y]. So the complex (T G, d) and its homology
are defined for more general objects than Lie algebras, for the Leibniz algebras.”
Knots and distributive homology
32
V.Lebed formalized this “linearization” in the case V has a central element 1 that is [x, 1] = 0[1, x] we color crossing as fola
b
lows:
a
b
b
a
. This led Lebed to describe face map
+
1
[a,b]
for chain complex of BLL algebras as in Figure 7.1; compare
the figure with Definition 7.2.
(2) A BLL module M over V is a k-module with a bilinear action
(still denoted [−, −] : M × V → M, satisfying the formula from
(1) for any x ∈ M and y, z ∈ V .
In a special case of M = k (k a ring with identity), the map
[−, −] : M × V → M is replaced by map ǫ : V → k which is
zero on commutators (Lie character), [Leb-2]).
( *)
d i (in BLL chain complex)
di
Y
X
X
X
i
M
V
V
M
V
i
V
V
+
[m,x 0 ]
x0
x i−1 x i+1
M
V
V
V
xn
V
i
i
+...+
m [x 0,x 1]
x i−1 x i+1
xn
m
x 0 [x ,x ] x
i−1 i
i+1
Figure 7.1; Comparing face map di in distributive and unital BLL algebra
Homology of (V, M) was constructed by Loday based partially on the
work of C.Cuvier [Cuv, Lod-1], they were unaware of the earlier work
by Bloh [Blo-2].
Definition 7.2. Let V be a BLL algebra, and M a BLL module over
V.
(1) We define Cn = M ⊗ V ⊗n+1 , that is C∗ = T V (tensor algebra).
For 0 ≤ i ≤ n, di : Cn → Cn−1 is given by:
X
di (x−1 , x0 , ..., xn ) =
(x−1 , x0 , ..., xj−1, [xj , xi ], xj+1 , xi−1 , xi+1 , ..., xn ).
j:−1≤j<i
If we take ΛV , the exterior algebra in place of T V . We will
be in the classical case of homology developed for Lie algebras
by Chevalley and Eilenberg [Ch-E].
(2) Assume that V is a split unital BLL algebra (that is V = V ′ ⊕k1,
and M = k, [1, x] = ǫ(x) (1 ∈ k x ∈ V ), then the “primitive”
xn
33
degenerate maps spi : Cn → Cn+1 is defined by
(x−1 , x0 , ..., xi−1 , (1, xi ) + (xi , 1), xi+1 , ..., xn )
p
si (x−1 , x0 , ..., xn ) =
(x−1 , x0 , ..., xi−1 , 1, 1, xi+1, ..., xn )
if xi ∈ V ′
if xi = 1
(3) Assume V is a free k-module with basis X and define degenerate
maps sgi : Cn → Cn+1 on X by doubling ith coordinate:
sgi (m, x0 , .., xn ) = (m, x0 , .., xi−1 , xi , xi , xi+1 , ..., xn ).
Lemma 7.3.
(1) (Cn (V, M), di ) is a presimplicial module, that is
di dj = dj−1 di for 0 ≤ i < j ≤ n.
(2) (Cn (X, M), di , spi ) is a weak simplicial module, and its homology
and degenerated homology are annihilated by ǫ(1).
(3) (Cn (X, M), di , sgi ) satisfies conditions (1) (2) and partially (3)
(di sj = sj−1di for i < j) of a simplicial module. Condition (4’),
di si − di+1 si = 0 holds iff [x, x] for all x ∈ X. The condition
(3) of a simplicial module for i > j + 1 requires the following
equality:
[xj , xi ] ⊗ [xj , xi ] = [xj , xi ] ⊗ xj + xj ⊗ [xj , xi ]
for basic elements (xj , xi ) ∈ X 2 for i > j + 1. Thus if we take
C∗ = V T /I divided by an ideal containing the above equation,
then (Cn (X, M), di , sgi ) is a very weak simplicial module. It is
a weak simplicial module iff additionally [x, x] = 0, for all x ∈
X.24
We leave the proof as an exercise for the reader however we make the
calculation in two small but typical cases which show that our axioms
are needed:
(i) Comparison of d0 d1 with d0 d0 (they should be equal):
d0 d1 (m; x0 , x1 ) = d0 (([m, x1 ]; x0 )+(m; [x0 , x1 ])) = ([m, x1 ], x0 )+[m, [x0 , x1 ]],
d0 d0 (m; x0 , x1 ) = d0 ([m, x0 ]; x1 )) = [[m, x0 )], x1 ],
Thus d0 d1 = d0 d0 if and only if ([m, x1 ], x0 ] + [m, [x0 , x1 ] = [[m, x0 ], x1 ]
which is the axiom of BLL-module.
(ii) Comparison of d1 d2 with d1 d1 (they should be equal):
d1 d2 (m; x0 , x1 , x2 ) = d1 (([m, x2 ]; x0 , x1 )+(m; [x0 , x2 ], x1 )+(m; x0 , [x1 , x2 ])) =
([[m, x2 ], x1 ]; x0 ) + (([m, x2 ]; [x0 , x1 ])+
([m, x1 ]; [x0 , x2 ]) + (m; [[x0 , x2 ], x1 ])+
([m, [x1 , x2 ]]; x0 ) + (m; [x0 , [x1 , x2 ]]),
and
d1 d1 (m; x0 , x1 , x2 ) = d1 ([m, x1 ]; x0 , x2 ) + (m; [x0 , x1 ], x2 )) =
24Notice
that in exterior algebra ΛV , the equation [xj , xi ] ⊗ [xj , xi ] = [xj , xi ] ⊗
xj + xj ⊗ [xj , xi ] holds (0 = 0), but in that case our degenerate map sgi would be a
zero map.
34
Knots and distributive homology
([[m, x1 ], x2 ]; x0 ) + ([m, x1 ]; [x0 , x2 ])+
([m, x2 ]; [x0 , x1 ]) + (m; [[x0 , x1 ], x2 ]).
Thus d1 d2 = d1 d1 if and only if the following sum is equal to zero:
([[m, x2 ], x1 ]; x0 ) + ([m, [x1 , x2 ]]; x0 ) − ([[m, x1 ], x2 ]; x0 )+
(m; [[x0 , x2 ], x1 ]) + (m; [x0 , [x1 , x2 ]]) − (m; [[x0 , x1 ], x2 ]).
The first part is equal to zero iff M is BLL-module and the second
part is equal to zero iff V is BLL-algebra.
Remark 7.4. Lie algebra homology, as proved by Cartan and Eilenberg [Ca-E] can be obtained from homology of the universal enveloping
algebra UV = T V /(a ⊗ b − b ⊗ a = [a, b]) of the Lie algebra V . One
hopes for a similar connection between distributive homology and homology of the group associated to a wrack or quandle. One hint in this
direction is that in every group the following “distributivity” holds:
[[x, y −1 ], z]y [[y, z −1 ], x]z [[z, x−1 ], y]x = 1,
where [x, y] = x−1 y −1xy and xy = y −1 xy. This leads to the graded Lie
algebra associated to the group, via lower central series of the group
[Va].
8. Semigroup extensions and shelf extensions
The theory of extension of structures and related cocycles started
from two important examples from group theory:
(i) The extension of P SLn (C) by SLn (C) by I.Schur (1904), with related short exact sequence of groups [B-T]
0 → Z2 → SLn (C) → P SLn (C) → 1 and
the study of crystallographic groups Γ where we consider a short exact
sequence
0 → Zn → Γ → Γ/Zn → 1.
An extension of a group X by a group N is a short exact sequence of
groups
i
π
1→N →E→X→1
(some people call this an extension of N by X [B-T, Bro, Ma-Bi]).
Consider a set-theoretic section s : X → E (that is πs = IdX ). Every element of E is a unique product as(x) for a ∈ N and x ∈ X
(coset decomposition), thus we have E = N × X as sets (here e →
(es(π(e−1 )), π(e)) and es(π(e−1 ))(sπ(e)) = e as needed. The inverse
map is (a, x) → as(x).
This motivates study of extension of magmas as study of projections
π : A × X → X with various structures preserved. In particular,
35
we compare semigroup extension of a semigroup by an abelian group
with the shelf extension of a shelf by an Alexander quandle. We start
from a general concept of a dynamic cocycle in a magma case and
then in associative and distributive cases and in both we relate to the
(co)homology of our structures. Extension of modules, groups and Lie
algebras is described in the classical book by Cartan and Eilenberg
[Ca-E], distributive case was developed in [CES-2, A-G, CKS].
Definition 8.1. Let (X; ∗) be a magma, A a set, and π : A × X → X
the projection to the second coordinate. Any magma structure on A×X
for which π is an epimorphism, can be given by a system of functions
φa1 ,a2 (x1 , x2 ) : X × X → A by:
(a1 , x1 ) ∗ (a2 , x2 ) = (φa1 ,a2 (x1 , x2 ), x1 ∗ x2 ).
Functions φa1 ,a2 (x1 , x2 ) are uniquely defined by the multiplication on
A×X, thus binary operations on A×X agreeing with π are in bijection
with choices of functions φa1 ,a2 . If we require some special structure
on (X; ∗) (e.g. associativity or right-distributivity) we obtain some
property of φa1 ,a2 (x1 , x2 ) which we call a dynamical co-cycle property
for the structure.
(1) Let (X; ∗) be a semigroup; in order that an action on A × X is
associative we need:
((a1 , x1 ) ∗ (a2 , x2 )) ∗ (a3 , x3 ) = (φa1 ,a2 (x1 , x2 ), x1 ∗ x2 ) ∗ (a3 , x3 ) =
(φφa1 ,a2 (x1 ,x2),a3 (x1 ∗ x2 , x3 ), (x1 ∗ x2 ) ∗ x3 )
to be equal to
(a1 , x1 ) ∗ ((a2 , x2 )) ∗ (a2 , x3 )) = (a1 , x1 ) ∗ (φa2 ,a3 (x2 , x3 ), x2 ∗ x3 ) =
(φa1 ,φa2 ,a3 (x2 , x3 ))(x1 , x2 ∗ x3 ), x1 ∗ (x2 ∗ x3 )).
Thus the dynamical cocycle condition in the associative case has
a form:
(φφa1,a2 (x1 ,x2 ),a3 (x1 ∗ x2 , x3 ) = φa1 ,φa2 ,a3 (x2 ,x3 ) (x1 , x2 ∗ x3 )).
(2) Let (X; ∗) be a shelf; in order that an action on A × X is right
self-distributive we need:
((a1 , x1 ) ∗ (a2 , x2 )) ∗ (a2 , x3 ) = (φa1 ,a2 (x1 , x2 ), x1 ∗ x2 ) ∗ (a3 , x3 ) =
(φφa1 ,a2 (x1 ,x2 ),a3 ) (x1 ∗ x2 , x3 ), (x1 ∗ x2 ) ∗ x3 )
to be equal to
((a1 , x1 ) ∗ (a3 , x3 )) ∗ ((a2 , x2 ) ∗ (a3 , x3 )) =
(φa1 ,a3 (x1 , x3 ), x1 ∗ x3 ) ∗ (φa2 ,a3 (x2 , x3 ), x2 ∗ x3 ) =
(φφa1 ,a3 (x1 ,x3 ),φa2 ,a3 (x2 ,x3 ) (x1 ∗ x3 , x2 ∗ x3 ), (x1 ∗ x3 ) ∗ (x2 ∗ x3 )).
36
Knots and distributive homology
Thus the dynamical cocycle condition in right-distributive case
has a form:
φφa1 ,a2 (x1 ,x2 ),a3 (x1 ∗ x2 , x3 ) = φφa1 ,a3 (x1 ,x3 ),φa2 ,a3 (x2 ,x3 ) (x1 ∗ x3 , x2 ∗ x3 ).
(3) We assume now that (X; ∗) is an entropic (inner turn) magma,
that is (a ∗ b) ∗ (c ∗ d) = (a ∗ c) ∗ (b ∗ d) for any a, b, c, d ∈ X.
We look for condition on the dynamical co-cycle so that A × X
is entropic. We need
((a1 , x1 ) ∗ (a2 , x2 )) ∗ ((a3 , x3 ) ∗ (a4 , x4 )) =
(φa1 ,a2 (x1 , x2 ), x1 ∗ x2 ) ∗ (φa3 ,a4 (x3 , x4 ), x3 ∗ x4 ) =
(φφa1,a2 (x1 ,x2 ),φa3 ,a4 (x3 ,x4) (x1 ∗ x2 , x3 ∗ x4 ), (x1 ∗ x2 ) ∗ (x3 ∗ x4 ))
to be equal to
((a1 , x1 ) ∗ (a3 , x3 )) ∗ ((a2 , x2 ) ∗ (a3 , x3 )) =
(φa1 ,a3 (x1 , x3 ), x1 ∗ x3 ) ∗ (φa2 ,a4 (x2 , x4 ), x2 ∗ x4 ) =
(φφa1 ,a3 (x1 ,x3 ),φa2 ,a4 (x2 ,x4 ) (x1 ∗ x3 , x2 ∗ x4 ), (x1 ∗ x3 ) ∗ (x2 ∗ x4 )).
Thus the dynamic cocycle condition in entropic case has the
form:
φφa1,a2 (x1 ,x2),φa3 ,a4 (x3 ,x4 ) (x1 ∗x2 , x3 ∗x4 ) = φφa1 ,a3 (x1 ,x3 ),φa2 ,a4 (x2 ,x4 ) (x1 ∗x3 , x2 ∗x4 ).
We illustrate the above by several examples, starting from a classical
group extension by an abelian group. Consider the extension E of a
group X by an abelian group A; this is described by a short exact
sequence of groups:
π
0→A→E→X →1
As noted before, E = A × X as a set and bijection depends on a
section s : X → E. Furthermore X acts on A (we have X × A → A)
and the action is given by conjugation: x(a) = s(x)a(s(x)−1 and does
not depend on the choice of s as A is commutative).
For a semigroup this is the starting point.
Let X be a semigroup, π : A × X a projection and a semigroup X act
on a set A. We define a product on A × X by the formula:
(a1 , x1 )(a2 , x2 ) = (a1 + x1 (a2 ) + f (x1 , x2 ), x1 x2 ).
The function f : X × X → A, as in the group case, arise by comparing section of a multiplication with multiplication of sections, that
is s(x1 x2 ) = f (x1 , x2 )s(x1 )s(x2 ). We assume that the action x :
A → A is a group homomorphism for any x and it is associative
(x1 (x2 (a) = (x1 x2 )(a)). The associativity of the product on A × X
37
is equivalent to the condition on f : X × X → A of the form25:
x1 (f (x2 , x3 )) − f (x1 x2 , x3 ) + f (x1 , x2 x3 ) − f (x1 , x2 ) = 0 which we call
a second cocycle condition (relation to homology of groups defined before, will be explained). Thus φa1 ,a2 (x1 , x2 ) = a1 + x1 (a2 ) + f (x1 , x2 ),
is an example of a dynamical cocycle for an associative structure. We
should stress that for a semigroup there may be choice for a dynamical
cocycle but for a group it is unique (see e.g. [Bro]). f : X × X → A
is a cocycle for a chain complex introduced in Definition 4.1 for the
trivial action (and generally Definition 4.2); we have:
∂ 2 (f )(x1 , x2 , x3 ) =
f (∂2 ((x1 , x2 , x3 )) = f ((x2 , x3 ) − (x1 x2 , x3 ) + (x1 , x2 x3 ) − (x1 , x2 )) =
f (x2 , x3 ) − (x1 x2 , x3 ) + f (x1 , x2 x3 ) − f (x1 , x2 ) = 0.
If action of X on A is not necessarily trivial, we define cohomology
H n (G, C) with a cochain complex Cn = Hom(ZGn → A) and
∂ n : C n → C n+1 is given by ∂ n (f )(x1 , ..., xn , xn+1 ) =
n
X
x1 f (x2 , ..., xn+1 )+ (−1)i f (x1 , ..., xi xi+1 , ...xn+1 )+(−1)n+1 (x1 , ..., xn ).
i=1
8.1. Extensions in right distributive case. We give here two examples of extension in right distributive case:
I. Let (X; ∗) be a shelf and A an abelian group with a given homomorphism t : A → A (equivalently, A is a Z[t] module). We define a shelf
structure (called Alexander extension [CES-2, CKS]) on A × X by the
formula:
(a1 , x1 )(a2 , x2 ) = (ta1 + (1 − t)a2 + f (x1 , x2 ), x1 ∗ x2 )
and right distributivity is equivalent to the condition on f : X ×X → A
which satisfies twisted cocycle condition:
t(f (x2 , x3 )−f (x1 , x3 )+f (x1 , x2 ))−f (x2 , x3 )+f (x1 ∗x2 , x3 )−f (x1 ∗x3 , x2 ∗x3 ) = 0.
25Calculation
is as follows: Associativity,
((a1 , x1 )(a2 , x2 ))(a3 , x3 ) = (a1 , x1 )((a2 , x2 ))(a3 , x3 )) gives, after expanding each side:
((a1 , x1 )(a2 , x2 ))(a3 , x3 ) = (a1 + x1 (a2 ) + f (x1 , x2 ), x1 x2 )(a3 , x3 ) =
(a1 + x1 (a2 ) + f (x1 , x2 ) + (x1 x2 )(a3 ) + f (x1 x2 , x3 ), (x1 x2 )x3 ) and
(a1 , x1 )((a2 , x2 ))(a3 , x3 )) = (a1 , x1 )(a2 + x2 (a3 ) + f (x2 , x3 ), x2 x3 ) =
(a1 + x1 (a2 + x2 (a3 ) + f (x2 , x3 ) + f (x1 , x2 x3 ), x1 (x2 x3 ))
thus the associativity reduces to:
f (x1 , x2 )+f (x1 x2 , x3 ) = x1 (f (x2 , x3 ))+f (x1 , x2 x3 ) which is our 2-cocycle condition.
38
Knots and distributive homology
The calculation is as follows: Right self-distributivity
((a1 , x1 ) ∗ (a2 , x2 )) ∗ (a3 , x3 ) = ((a1 , x1 ) ∗ (a3 , x3 )) ∗ ((a2 , x2 ) ∗ (a3 , x3 ))
gives, after expanding each side:
((a1 , x1 )∗(a2 , x2 ))∗(a3 , x3 ) = (ta1 +(1−t)a2 +f (x1 , x2 ), x1 ∗x2 )∗(a3 , x3 ) =
(t(ta1 +(1−t)a2 +f (x1 , x2 ))+(1−t)a3 +f (x1 ∗x2 , x3 ), (x1 ∗x2 )∗x3 ) and
((a1 , x1 ) ∗ (a3 , x3 )) ∗ ((a2 , x2 ) ∗ (a3 , x3 )) =
(ta1 +(1−t)a3 +f (x1 , x3 ), x1 ∗x3 )∗(ta2 +(1−t)a3 +f (x2 , x3 ), x2 ∗x3 ) =
(t(ta1 + (1 − t)a3 + f (x1 , x3 )) + (1 − t)(ta2 + (1 − t)a3 +
f (x2 , x3 )) + f (x1 ∗ x3 , x2 ∗ x3 ), (x1 ∗ x3 ) ∗ (x2 ∗ x3 ))
This is equivalent to:
tf (x1 , x2 )+f (x1 ∗x2 , x3 ) = tf (x1 , x3 )+(1−t)f (x2 , x3 ))+f (x1 ∗x3 , x2 ∗x3 ),
and further to a cocycle in a (twisted) rack homology:
(∂ R f )(x1 , x2 , x3 ) =
−t(f (x2 , x3 ) − f (x1 , x3 )) + f (x1 , x2 ))+
f (x2 , x3 ) − f (x1 ∗ x2 , x3 ) + f (x1 ∗ x3 , x2 ∗ x3 ) = 0.
If there are two right self-distributive binary operations, ∗1 and ∗2 on
A×X represented by f1 and f2 respectively (that is (a1 , x1 )∗i (a2 , x2 ) =
(a1 ∗ a2 + fi (x1 , x2 ), x1 ∗ x2 , i = 1, 2), and there is a homomorphism
H : A×X → A×X given by H(a, x) = (a+c(x), x) for some c : X → A
then the homomorphism condition
H((a1 , x1 ) ∗1 (a2 , x2 )) = H(a1 , x1 ) ∗2 H(a2 , x2 ) is equivalent to
(a1 ∗ a2 + f1 (x1 , x2 ) + c(x1 ∗ x2 ), x1 ∗ x2 ) =
((a1 + c(x1 )) ∗ (a2 + c(x2 )) + f2 (x1 , x2 ), x1 ∗ x2 ) thus
ta1 +(1−t)a2 +c(x1 ∗x2 )+f1 (x1 , x2 ) = t(a1 +c(x1 )+(1−t)(a2 +c(x2 ))+
f2 (x1 , x2 ) so f1 (x1 , x2 ) − f2 (x1 , x2 ) = tc(x1 ) + (1 − t)c(x2 ) − c(x1 x2 ) =
(∂c)(x1 , x2 ). We can say that the second cohomology (here (twisted)
rack cohomology) H 2 (X, A) describes (shelf) extensions of X by A
of type described above, modulo described above equivalence, for an
abelian group A.
The dynamical cocycle is given by φa1 ,a2 (x1 , x2 ) = ta1 + (1 − t)a2 +
f (x1 , x2 ), [CKS].
II. Another family of extensions is given by the hull construction for
a multi-shelf (multi-RD-system) of Patrick Dehornoy and David Larue
[Deh-2, Lar], and its “G-group” generalization (which we call a twisted
hull) by Ishii, Iwakiri, Jang, and Oshiro [IIJO].
For a hull construction we need a distributive set of binary operations
39
on A indexed by elements of X, that is (a ∗x b) ∗y c = (a ∗y c) ∗x (b ∗y c),
and the “hull” shelf structure on A × X is given by:
(a1 , x1 ) ∗ (a2 , x2 ) = (a1 ∗x2 a2 , x1 ).
To see our construction as obtained from a dynamical cocycle we put
trivial operation on X (x ∗ y = x), and the dynamical cocycle is given
by φa1 ,a2 (x1 , x2 ) = a1 ∗x2 a2 .
Remark 8.2. If A = F (X) is a fee group on free generators X, then
the hull A × X related to the distributive set of operations ∗x on A
given by a1 ∗x a2 = a1 a−1
2 xa2 is a free rack generated by X (denoted
by F R(X)) as defined by Fenn and Rourke [F-R] (see also [CKS]). To
summarize, we have then F R(X) = F (X) × X with (a1 , x1 ) ∗ (a2 , x2 ) =
(a1 ∗x2 a2 , x1 ) = (a1 a−1
2 x2 a2 , x1 ).
The G-group generalization of hull to twisted hull, relaxes condition
that X is indexing distributive set of operation and we allow “twisted
distributivity”. That is: (a ∗x b) ∗y c = (a ∗y c) ∗x∗y (b ∗y c). Thus X
indexes operations on A satisfying “twisted distributivity”. In this case
the shelf structure on A × X is given by:
(a1 , x1 ) ∗ (a2 , x2 ) = (a1 ∗x2 a2 , x1 ∗ x2 ).
The fundamental example leading to “twisted distributivity” was already given by Joyce: Let G be a group and X be a subgroup of
hom(G, G). Then we define g1 ∗x g2 = x(g1 g2−1 )g2 and we get:
(g1 ∗x2 g2 ) ∗x3 g3 = (g1 ∗x3 g3 ) ∗x2 ∗x3 (g2 ∗x3 g3 )
where x2 ∗ x3 = x3 x2 x−1
3 .
Twisted distributivity is illustrated in Figure 8.1 below:
Knots and distributive homology
40
*
b
a
y *z
z
c
(b* z c) *y
y
g
b* z c
x
z
y
b
a
(b* z c)
b* z c
c
R
c
*z
3
(x * y) * z
a* y b
x* y
(a* y b) * z c
b* z c
x
c
Figure 8.1; Twisted distributivity
8.2. Extensions in entropic case. Let (X; ∗) be an entropic magma,
that is ∗ satisfies the entropic identity: (a ∗ b) ∗ (c ∗ d) = (a ∗ c) ∗ (b ∗
d). Let also A be an abelian group with a given pair of commuting
homomorphisms t, s : A → A and a constant a0 ∈ A; we consider
(A; ∗) as an entropic magma with an affine action a ∗ b = ta + sb + a0 .
Then we define a binary operation on A×X by (a1 ∗x1 )∗(a2 ∗x2 ) = (a1 ∗
a2 +f (x1 , x2 ), x1 ∗x2 ). In order for A×X to be entropic magma we need
entropic condition, or equivalently φa1 ,a2 (x1 , x2 ) = a1 ∗ a2 + f (x1 , x2 )
should be and entropic dynamic cocycle. This leads to entropic cocycle
condition:
tf (x1 , x2 )−tf (x1 , x3 )+sf (x3 , x4 )−sf (x2 , x4 )+f (x1 ∗x2 , x3 ∗x4 )−f (x1 ∗x3 , x2 ∗x4 ) = 0.26
26Calculation
is as follows: Entropic condition, ((a1 , x1 ) ∗ (a2 , x2 )) ∗ ((a3 , x3 ) ∗
(a4 , x4 )) = ((a1 , x1 ) ∗ (a3 , x3 )) ∗ ((a2 , x2 ) ∗ (a4 , x4 )) gives, after expanding each side:
((a1 , x1 ) ∗ (a2 , x2 )) ∗ ((a3 , x3 ) ∗ (a4 , x4 )) = (a1 ∗ a2 + f (x1 , x2 ), x1 ∗ x2 ) ∗ (a3 ∗ a4 +
f (x3 , x4 ), x3 ∗ x4 ) = ((a1 ∗ a2 ) ∗ (a3 ∗ a4 ) + tf (x1 , x2 ) + sf (x3 , x4 ) + f (x1 ∗ x2 , x3 ∗
x4 ), (x1 ∗ x2 ) ∗ (x3 ∗ x4 ))
Similarly ((a1 , x1 )∗ (a3 , x3 ))∗ ((a2 , x2 )∗ (a4 , x4 )) = ((a1 ∗ a3 )∗ (a2 ∗ a4 )+ tf (x1 , x3 )+
sf (x2 , x4 )+f (x1 ∗x3 , x2 ∗x4 ), (x1 ∗x3 )∗(x2 ∗x4 )), which reduces to (entropic) cocycle
condition tf (x1 , x2 ) + sf (x3 , x4 ) + f (x1 ∗ x2 , x3 ∗ x4 ) = tf (x1 , x3 ) + sf (x2 , x4 ) +
(x1 ∗ x3 , x2 ∗ x4 ).
41
The above formula may serve as a hint how to define (co)homology
in entropic case [N-P-4].
In particular ∂ : RX 4 → RX 2 may be given by:
∂(x1 , x2 , x3 , x4 ) =
t(x1 , x2 )−t(x1 , x3 )+s(x3 , x4 )−s(x2 , x4 )+(x1 ∗x2 , x3 ∗x4 )−(x1 ∗x3 , x2 ∗x4 )
and ∂ : RX 2 → RX may be given by: ∂(x1 , x2 ) = tx1 − x1 ∗ x2 + sx2
(here it agrees with the rack case for s = 1 − t).
Remark 8.3. An important, but not yet fully utilized, observation from
[N-P-4], is that we can consider atomic boundary functions
∂ (∗) ((x1 , x2 , x3 , x4 ) = (x1 ∗x2 , x3 ∗x4 )−(x1 ∗x3 , x2 ∗x4 ) and ∂ (∗) (x1 , x2 ) =
−x1 ∗ x2 , and consider also the left trivial binary operations x ∗0 y = x,
and the right trivial binary operations x ∗∼ y = y and then to recover
∂ as a three term entropic boundary function, for the multi-entropic
system (∗, ∗0 , ∗∼ ), by the formula ∂ = ∂ (∗) − t∂ (∗0 ) − s∂ (∗∼ ) .
9. Degeneracy for a weak and very weak simplicial
module
We expand here on Subsections 3.2 and 3.3 and discuss degenerate
part of distributive homology in the general context of weak and very
weak simplicial modules.
Quandle homology is build in analogy to group homology or Hochschild
homology of associate structures. In the unital associative case we deal
with simplicial sets (or modules) and it is a classical result of Eilenberg
and Mac Lane that the degenerate part of a chain complex is acyclic
so homology and normalized homology are isomorphic (see Subsection
3.2). It is not the case for distributive structures, e.g. for quandles
or spindles. Quandle homology or even one term distributive homology of spindles may have nontrivial degenerate part. The underlining
homological algebra structure is a weak simplicial module and in this
case the degenerate part is not necessarily acyclic and the best one
can say is that the degenerate part has a natural filtration so yields
a spectral sequence which can be used to study degenerate homology.
In the concrete case of quandle homology (motivated by and applicable to knot theory) it is proven that the homology (called the rack
homology) splits into degenerate and normalized (called the quandle
homology) parts [L-N]. Otherwise no clear general connection between
degenerate and quandle part were observed. We prove in the joint paper with Krzysztof Putyra [Pr-Pu-2] that the degenerated homology
of a quandle is fully determined by quandle homology via a Künneth
type formula.
42
Knots and distributive homology
10. Degeneracy for a weak simplicial module
Here we give a few general observations about degenerate part of
a weak simplicial module. They are related to concrete work in the
distributive case done in [Pr-Pu-2].
Consider a weak simplicial module (Cn , di , si ) (see Subsection 3.2
and [Prz-5]). As checked in Corollary 3.4, the filtration by degenerate
elements Fnp = span(s
P0 (Cn−1 ), ..., sp (Cn−1 )) is preserved by the boundary operation ∂n = ni=0 (−1)i di . In Subsection 3.3 we constructed a
0
degenerate bicomplex (Ei,j
, dv , dh )
We discuss here the fact that a weak simplicial complex has also dual
filtration (or better to say it has left and write filtrations). We define
the dual (or opposite) filtration F̂np = span(sn−1 (Cn−1 ), ..., sn−p (Cn−1)).
We start our dual description from a presimplicial module:
If (Cn ; di ) is a presimplicial module then we define dˆi = dn−i and
notice that (Cn ; dˆi ) is also a presimplicial module with ∂ˆn = (−1)n ∂n
and unchanged homology. More generally we have:
Proposition 10.1.
(i) If (Cn ; di ) is a presimplicial module then (Cn ; dˆi ) is also a presimplicial module.
(ii) If si : Cn → Cn+1, 0 ≤ i ≤ n are degenerate map, define ŝi = sn−i .
Then if (Cn ; di , si ) is a (weak, or very weak) simplicial module then
(Cn ; dˆi , ŝi ) is also a a (weak or very weak) simplicial module.
Proof. (i) For i < j we have n − j < n − i, so:
(1)
dˆi dˆj = dˆi dn−j = dn−1−i dn−j = dn−j dn−i = dˆj−1 dˆi .
(ii) For a better presentation let us list conditions of a simplicial module
for (Cn , dˆi , ŝi ), one by one:
(1̂) dˆi dˆj = dˆj−1dˆi f or i < j.
(2̂) ŝi ŝj = ŝj+1 ŝi , 0 ≤ i ≤ j ≤ n,
ŝj−1dˆi if i < j
(3̂) dˆi ŝj =
ŝj dˆi−1 if i > j + 1
(4̂′ ) dˆi ŝi = dˆi+1 ŝi .
(4̂) dˆi ŝi = dˆi+1 ŝi = IdMn .
Proposition 10.1 follows from the following lemma.
Lemma 10.2. Consider (Cn ; di , si ) and its complementary (dual) (Cn ; dˆi , ŝi ),
then conditions (x) and (x̂) are equivalent
43
Proof. Equivalence of (1) and (1̂) was already established. Other parts
are equally simple but we prove them for completeness:
(2) ⇔ (2̂) (we assume i ≤ j or equivalently n − j ≤ n − i):
(2)
ŝi ŝj = ŝi sn−j = sn+1−i sn−j = sn−j sn−i = ŝj+1 ŝi .
(3) ⇔ (3̂) First assume that i < j (i.e. n + 1 − i > n − j + 1) then:
(3)
dˆi ŝj = dn+1−i sn−j = sn−j dn−i = ŝj−1 dˆi .
Second assume that i > j + 1 (i.e. n + 1 − i < n − j), then:
(3)
dˆi ŝj = dn+1−i sn−j = sn−j−1dn−i+1 = ŝj dˆi−1 .
(4′ ) ⇔ (4̂′ ). We have:
′
(4 )
dˆi ŝi = dn+1−i sn−i = dn−i sn−i = dˆi+1 ŝi .
(4) ⇔ (4̂). We have:
(4)
dˆi ŝi = dn+1−i sn−i = Id.
Remark 10.3. If Cn = ZX n+1 then we can consider the map Iˆ :
ˆ 0 , x1 , ..., xn ) = (xn , ..., x1 , x0 ) (or suc(Cn ; di ) → (Cn ; dˆi ) given by I(x
ˆ
cinctly I(x) = x̂)).
Our results ( Proposition 10.1 and Lemma 10.2 hold for very weak
simplicial modules, weak simplicial modules, and simplicial modules.
In particular, for a weak simplicial module the dual filtration of CnD ,
F̂np = span(ŝ0 (Cn−1 ), ŝ1 (Cn−1 ), ..., ŝp (Cn−1 ) leads to a spectral sequence
and a bicomplex. Here we give a few general remarks to summarize
basic facts:
A weak simplicial module yields two filtrations: Fnp and the dual (complementary) one F̂np . By the definition we have
n
n
X
X
i
n
∂n =
(−1) di = (−1)
(−1)i dˆi = (−1)n ∂ˆn
i=0
i=0
Furthermore, on sp (Cn−1 ) we have:
∂n sp =
n
X
i=0
i
(−1) di sp =
p−1
X
i=0
i
p
(−1) di sp +(−1) dp sp +(−1)
p+1
dp+1 sp +
n
X
(4′ )
(−1)i di sp =
i=p+2
Knots and distributive homology
44
p−1
X
i
(−1) di sp +
i=0
p−1
X
(3)
(−1)i di sp =
i=p+2
i
(−1) sp−1 di +
i=0
Pp−1
n
X
n
X
(−1)i sp di−1 .
i=p+2
i
Clearly i=0 (−1) sp−1di (Cn−1 ) belongs to Fnp−1.
0
The formulas above lead to the bicomplex with Ep,q
= Mp,q =
Pn
P
p−1
i
v
p
p−1
h
Fn /Fn , where n = p+q, d = i=0 (−1) di sp and d = i=p+2 (−1)i di sp .
The equality dh dh = 0 = dv dv and dh dv = −dv dh follows directly from
the weak simplicial module structure.
If we replace the filtration Fnp by F̂np we see that the spectral sequence
is modified; it is analogous, but not the same, as when dh is replaced
by dv .
Remark 10.4. An acute observer will notice immediately27 that we
deal not only with a bicomplex but also with pre-bisimplicial category
(set or module). For completeness I recall definitions after [Lod-1],
page 459:
We define a bisimplicial object but in a same vain we can define prebisimplicial category, and weak bisimplicial category: “ By definition a
bisimplicial object in a category C is a functor
X : ∆op × ∆op → C.
Such a bisimplicial object can be described equivalently by a family of
objects Mp,q , p ≥ 0, q ≥ 0, together with horizontal and vertical faces
and degeneracies:
dhi : Mp,q → Mp−1,q ,
shi : Mp,q → Mp+1,q , where 0 ≤ i ≤ p
dvi : Mp,q → Mp,q−1 ,
svi : Mp,q → Mp,q+1 , where 0 ≤ i ≤ q
which satisfy the classical simplicial relations horizontally and vertically and such that horizontal and vertical operations commute. For
any bisimplicial set X there are three (homeomorphic) natural ways to
make geometric realization, |X| of X. Loday notes that any bisimplicial set X gives rise to the bisimplicial module RX and H∗ (|X|, R) =
H∗ (T ot(RX)), [Lod-1].
Example 10.5. A natural example of a (pre)-bisimplicial set or module
is obtained by a Cartesian (or tensor) product of (pre)-simplicial sets
(or modules). Namely:
27Victoria
Lebed studied this before me in context of her prebraided category.
45
(×) Let Mp,q = Cp ×Cq′ where (Cn , di ) and (Cn′ , d′i ) are (pre)simplicial
sets. We define dhi = di × IdCq′ and dvi = IdCp × d′i . In the case
(Cn , di , si ) and (Cn′ , d′i , s′i ) are (weak)-simplicial sets we get Mp,q
a (weak)-simplicial set with shi = si × IdCq′ and svi = IdCp × s′i .
(⊗) Let Mp,q = Cp ⊗Cq′ where (Cn , di ) and (Cn′ , d,i ) are (pre)simplicial
modules. Then Mp,q is a (pre)simplicial modules with dhi =
di ⊗ IdCq′ and dvi = IdCp ⊗ d′i . Similarly in the case (Cn , di , si )
and (Cn′ , d′i , s′i ) are (weak)-simplicial modules.
10.1. Right filtration of degenerate distributive elements.
We restrict ourselves here to a weak simplicial module yielded by a
distributive structures.
Let (X; ∗) be a spindle that is a magma which is right distributive
((a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c)) and idempotent (a ∗ a = a).
Definition 10.6.
(i) Let ŝi = sn−i : Cn → Cn+1 is given by
ŝi (x0 , x1 , ..., xn ) = (x0 , .., xn−i−1 , xn−i , xn−i , xn−i+1 , ..., xn ),
that is we double the letter on the position n − i (or i from the
end) if we count from zero.
(ii) We define F̂np = span(ŝ0 (Cn−1 ), ŝ1 (Cn−1 ), ..., ŝn−1 (Cn−1 )) in Cn (X).
F̂np form a boundary coherent filtration of Cn (X):
0 ⊂ F̂n0 ⊂ F̂n1 ⊂ F̂nn−1 = CnD .
n
(iii) Let Ĝrpn be the associated graded group: Ĝrpn = F̂np /F̂p−1
.
If, as before, we define face maps dˆi = dn−i then (Cn , dˆi , ŝi ) is a weak
simplicial module. Thus F̂np is a graded filtration ripe for the spectral
P
sequence. (We already noticed that ∂ˆn = ni=0 (−1)i dˆi = (−1)n ∂n .)
We consider the spectral sequence of the filtration starting from the
p
0
= Ĝrp+q
initial page Êp,q
. It is the first main observation of [Pr-Pu-2]
r
r
that the spectral sequence (Êp,q
, ∂ˆp,q
) stabilizes on the first page, and
eventually one term spindle homology can be computed easily from the
normalized part.
10.2. Integration maps ûi : F̂np → F̂np−1 .
r
The main tools to show that the spectral sequence Êp,q
stabilizes on the
first page are the maps (which we can call integration) ûi : F̂np → F̂np−1,
illustrated in Figure 10.1 below28, they serve to show that the right
28In
braid notation, ûi can be expressed as σp σp−1 ...σi+1 σi .
Knots and distributive homology
46
degenerated filtration spectral sequence has all dr (r > 0) trivial and
that homology splits.
p p
i 0
ui
p p
i 0
di
Figure 10.1; The maps ûi and dˆi
We check that the maps ûi satisfy:
(1) for i < p: dˆi (y) = dˆp+1ûi (y), where y ∈ ŝp (Cn−1 ).
(2) For p > i2 > i1 : dˆp+1ûi2 −1 ûi1 = dˆi1 ûi2 .
from this follows
(2’) For p > i2 > i1 : dˆp ûi2 −1 ûi1 = dˆi2 −1 ûi1 .
(2) and (2’) are illustrated in Figure 10.2 and 10.3 below.
(j) For p > ij > ij−1 > ... > i1 ≥ 0 one has dˆp+1 ûij −j+1...ûi2 −1 ûi1 = dˆi1 ûij −j+2 ...ûi3 −1 ûi2 .
From this follows:
dˆp+1−k+1ûij −j+1 ...ûik −k+1 ...ûi2 −1 ûi1 =
(j ≥ k)
dˆik −k+1 ûij −j+2 ...ûik+1 −(k+1)+2 ûik−1 −(k−1)+1 ...ûi2 −1 ûi1 .
xi xi
x p+1=x p
d
p+1
2
u i −1 u i
2
1
1
x0
x p+1=x p x i
di u i
Figure 10.2; Property (2): dˆp+1ûi2 −1 ûi1 = dˆi1 ûi2
1
2
2
xi x
0
1
47
xn
xi xi
x p+1=x p
d
p
2
u i −1 u i
2
1
x0
xn
x p+1=x p
d
1
xi xi
2
1
u
i −1 i 1
2
Figure 10.3; Property (2): dˆp ûi2 −1 ûi1 = dˆi2 −1 ûi1
Remark 10.7. We wrote formulas for dˆs ûik −k+1 ...ûi1 −1 ûi1 only in the
case of p + 1 − k ≤ s ≤ p + 1 as it is needed to compute the degenerate
part of one term distributive homology of a spindle.
10.3. Weak simplicial modules with integration. Here we formalize above equations to define weak simplicial module with integration
1
for which spectral sequence stabilizes on Ep,q
and homology splits.
We consider a weak simplicial module (Cn , di , si ) with an additional
structure, ui : sp (Cn−1 ) → sp−1 (Cn−1 ) for i < p, where maps ui satisfy
condition (j) for any j. This additional structure allows us to split
degenerate part.
Definition 10.8. We say that (Cn , di , si , ui ) is a weak simplicial module with integration if (Cn , di , si ) is a weak simplicial module, ui : Fnp →
Fnp−1 (0 ≤ i < p) and the following hold:
(1) dp+1 ui = di (i < p).
(j) dp+1 uij −j+1...ui2 −1 ui1 = di1 uij −j+2 ...ui3 −1 ui2 ,
where p > ij > ... > i1 ≥ 0.
A weak simplicial module with integration leads to bicomplex which
1
stabilizes on Ep,q
and eventually splits using the maps fnp : Fnp /Fnp−1 →
Lp
i
i−1
i=0 Fn /Fn .
11. Degeneracy for a very weak simplicial module
We have considered, previously, the degenerate subcomplex in the
case of a weak simplicial module, however if (Cn , di , si ) is only a very
x0
48
Knots and distributive homology
weak simplicial module, that is di si is not necessarily equal to di+1 si ,
we can still construct the analogue of a degenerate subcomplex (and
degenerate filtration (compare Remark. 3.4 in [Prz-5]).
Let C = (Cn , di , si ) be a very weak simplicial module, that is axioms
(1)-(3) of Definition 3.3 hold. We do not necessarily have the condition
di si − di+1 si is equal to zero so it is of interest to study an obstruction
to zero: ti = di si − di+1 si . We have;
Lemma 11.1. Let ti : Cn → Cn where ti = di si − di+1 si in a very weak
simplicial module, then:
(i)

 tj−1 di if i < j
0 if i = j
di tj =

tj di if i > j
(ii) ti tj = tj ti .
(iii) It follows from (i) that we have boundary preserving filtrations:
F0t = t0 (Cn ) ⊂ F1t = span(t0 (Cn ), t1 (Cn )) ⊂ ... ⊂ Fnt = span(t0 (Cn ), ..., tn (Cn )) = F t
(iii) We have also boundary preserving filtrations:
F0tD = span(t0 (Cn ), s0 (Cn−1 ) ⊂ ... ⊂
tD
Fn−1
Cn = span(t1 (Cn ), s1 (Cn−1 ), ..., tn−1 (Cn ), sn−1 (Cn−1 ) ⊂
FntD Cn = span(t1 (Cn ), s1 (Cn−1 ), ..., tn−1 (Cn ), sn−1 (Cn−1 ), tn (Cn )) = F tD .
Or the filtration
D
0 ⊂ (F0D + F t )/F t ⊂ ... ⊂ (Fn−1
+ F t )/F t = F tD /F t
(iv) (∂n tp − tp ∂n )(tp (Cn )) ⊂ tp−1 (Cn ). In particular, tp is a chain
t
map on Fpt (Cnt )/Fp−1
(Cnt ).
F tD is likely the best proxy of degenerated subchain complex so I
call it the generalized degenerated subchain complex of a very weak
simplicial module. The quotient Cn /FntD Cn is an analogue of a normalized chain complex (in quandle theory the quandle chain complex
for any distributive structure, not necessarily spindle or quandle).
Proof. (i) (i < j case): we have here, di tj = di dj sj − di dj+1 sj =
dj−1di sj − dj di sj = dj−1sj−1 di − dj sj−1di = tj−1 di ,
(i = j case)a we have, : dj tj = dj dj sj − dj dj+1sj = 0,
(i > j case): we have, di tj = di dj sj − di dj+1 sj = DO
(ii) First we show that:
(ii’) ti sj = sj ti for i < j we have, stressing which property is used:
ti sj = (di si − di+1 si )sj = (di − di+1 )si sj
(2)si sj =sj+1 si
=
(di − di+1 )sj+1 si =
49
(3)
di sj+1 si − di+1 sj+1si = sj di si − sj di+1 si = sj ti .
(Similarly we prove that ti sj = sj ti for i > j. Now we complete the
proof that ti tj = tj ti . We have, assuming i < j:
(ii′ )
(i)
ti tj = ti (dj sj − dj+1sj ) = dj ti sj − dj+1ti sj =
= dj sj ti − dj+1 sj ti = tj ti as needed.
(iii) We start from computing ∂n tp stressing each time which property
is used:
p−1
n
n
X
X
X
i
i
(−1)i di tp =
(−1) di tp + 0 +
∂n tp =
(−1) di tp =
i=0
i=1
i=p+1
p−1
n
X
X
i
(−1)i tp di
(−1) tp−1 di +
i=0
i=p+1
Fpn
This is proving that filtration
is boundary preserving and also that
we deal with bicomplex. Thus we can construct the spectral sequence
from the bicomplex.
Again if we work with a shelf and filtration from the right then
1
it seems to stabilize at Ep,q
and homology splits, like in the case of
right degenerate filtration of a spindle, F̂np . See Figure 11.1 below for
graphical interpretation of t̂i . ûti will be defined by analogy to ûi .
n
n
i 0
d i+1s i
ti
i
0
di s i
Figure 11.1; The maps t̂i as the difference of dˆi+1 ŝi and dˆi ŝi
We leave to a reader the development of these ideas, specially in the
case of multiterm distributive homology (compare [Pr-Pu-2]).
Knots and distributive homology
50
11.1. Introduction to t-simplicial objects. We can extract properties of maps ti to obtain a new version of a simplicial object which
we will call t-simplicial object.
Definition 11.2. Let C be a category and (Xn , di , ti ) the sequence of
objects Xn , n ≥ 0, and morphisms di : Xn → Xn−1 , ti : Xn → Xn , 0 ≤
i ≤ n . We say that (Xn , di , ti ) is a t-simplicial object if the following
four conditions hold (the first is the condition of a presimplicial object):
(1t ) di dj = dj−1 di for i < j.
(2t ) ti tj = tj ti .
(3t )
tj−1 di
di tj =
tj di
if i < j
if i > j
(4t ) di ti = 0.
P
Let (Cn , di , ti ) be a t-simplicial module, with ∂n = ni=0 (−1)i di then
(Cn , di , ti ) leads to a bicomplex (Cp,q , dh , dv ) where dh and dv are defined
up sign/shift by:
h
d =
p−1
X
i=0
i
v
(−1) di and d =
n
X
(−1)i di .
i=p+1
We also can define a t analogue of a bisimplicial object, which we
call a t-bisimplicial object, (Cp,q , dhi , dvj ), 0 ≤ i ≤ p, 0 ≤ j ≤ q, with
dhi = di and dvj = dp+1+i (however some adjustment is needed to have
p + q = n (or just p + q = n − 1).
Problem 11.3. The relation di ti = 0 is crucial. Can one find for it
general setting (say, ti as the marker for horizontal and vertical parts
of a bicomplex)? Other applications?
12. From distributive homology to Yang-Baxter
homology
We can extend the basic construction from the introduction, still
using very naive point of view, as follows: Fix a finite set X and color
semi-arcs of D (parts of D from a crossing to a crossing) by elements of
X allowing different weights from some ring k for every crossing (following statistical mechanics terminology we call these weights Boltzmann
weights). We allow also differentiating between a negative and a positive crossing; see Figure 12.1.
51
a
b
c
d
c
d
a
b
ab
Rc d
cd
Ra b
a,b
c,d
Figure 12.1; Boltzmann weights Rc,d
and R̄a,b
for positive and negative crossings
We can now generalize the number of colorings to state sum (basic
notion of statistical physics) by multiplying Boltzmann weight over all
crossings and adding over all colorings:
X
Y
a,b
col(X;BW ) (X) =
R̂c,d
(p)
φ∈colX (D) p∈{crossings}
a,b
a,b
a,b
where R̂c,d
is Rc,d
or R̄c,d
depending on whether p is a positive or
negative crossing (see [Jones]). Our state sum is an invariant of a
diagram but to get a link invariant we should test it on Reidemeister
moves. To get analogue of a shelf invariant we start from the third
Reidemeister move with all positive crossings. Here we notice that, in
analogy to distributivity, where passing through a positive crossing was
coded by a map R : X × X → X × X with R(a, b) = (b, a ∗ b). Thus in
the general case passing through a positive crossing is coded by a linear
map R : kX ⊗kX → kX ⊗kX and in basis X the map R is given by the
a,b
|X|2 × |X|2 matrix with entries (Rc,d
). The third Reidemeister move
leads to the equality of the following maps V ⊗ V ⊗ V → V ⊗ V ⊗ V
where V = kX:
(R ⊗ Id)(Id ⊗ R)(R ⊗ Id) = (Id ⊗ R)(R ⊗ Id)Id ⊗ R)
This is called the Yang-Baxter equation and R is called a pre-YangBaxter operator. If R is additionally invertible it is called a Yanga,b
Baxter operator. If entries of R−1 are equal to R̄c,d
then the state sum
is invariant under “parallel” (directly oriented) second Reidemeister
move.29
For a given pre-Yang-Baxter operator we attempt to find presimplicial module, from which homology will be derived.
29We
should stress that to find link invariants it suffices to use directly oriented
second and third Reidemeister moves in addition to both first Reidemeister moves,
as we can restrict ourselves to braids and use the Markov theorem. This point of
view was used in [Tur].
Knots and distributive homology
52
Figure 12.2 below illustrate various graphical interpretation of the
generating morphism di of the presimplicial category ∆op
pre . They are
related to homology of a set-theoretic Yang-Baxter equation of CarterKamada-Saito [CES-2] and Fenn [FIKM], and to homology of YangBaxter equation of Eisermann [Eis-1, Eis-2]. We should also acknowledge stimulating observations by Ivan Dynnikov.
i
Figure 12.2; Various interpretation of the graphical face map di
12.1. Graphical visualization of Yang-Baxter face maps. The
presimplicial set corresponding to (two term) Yang-Baxter homology
has the following visualization. In the case of a set-theoretic YangBaxter equation we recover the homology of [CES-2].
M0 V
i
V
i
V
Diagramatic interpretation of a face map d
Y−B
i
Figure 12.3; Graphical interpretation of the face map di
V M n+1
53
Our graphical model allows easy calculation:
Example 12.1. Assume R : X × X → X × X generates set-theoretic
Yang-Baxter operator with R(x, y) = (R1 (x, y), R2(x, y)). Then
∂ Y B (x1 , x2 , x3 , x4 ) = ∂ ℓ − ∂ r
∂ ℓ (x1 , x2 , x3 , x4 ) = ((x2 , x3 , x4 ) − (R2 (x1 , x2 ), x3 , x4 )+
(R2 (x1 , R1 (x2 , x3 ), R2 (x2 , x3 ), x4 )−(R2 (x1 , R1 (x2 , R1 (x3 , x4 ), R2 (x2 , R1 (x3 , x4 ), R2 (x3 , x4 ))
∂ r (x1 , x2 , x3 , x4 ) = (R1 (x1 , x2 ), R1 (R2 (x1 , x2 ), x3 ), R1 (R2 (R2 (x1 , x2 ), x3 ), x4 )−
(x1 , R1 (x2 , x3 ), R1 (R2 (x2 , x3 ), x4 ) + (x1 , x2 , R1 (x3 , x4 )) − (x1 , x2 , x3 ).
We have generally for any n:
∂nℓ
n
X
=
(−1)i−1 dℓi with
i=1
dℓi (x1 , ..., xn ) =
(R2 (x1 , R1 (x2 , R1 (x3 , ..., R1 (xi−1 , xi )))), ..., R1 (xi−1 , xi )), xi+1 , ..., xn ).
Similarly we have, directly from Figure 12.3, for any n:
∂nr
=
n
X
(−1)i−1 dri with
i=1
dri (x1 , ..., xn ) =
(x1 , ..., xi−1 , R1 (xi , xi+1 ), ..., R1 (R2 (R2 (...(R2 (xi , xi+1 ), xi+2 ), ..., xn−1 )xn )))).
13. Geometric realization of simplicial and cubic sets
A simplicial (or presimplicial) set (or space, more generally) can be
a treated as an instruction of how to glue a topological space from
pieces (simplexes in the most natural case). the result is CW complex
or more precisely a ∆−complex in the terminology used in [Hat]. That
is an object in Xn is a name/label for a an n dimensional simplex, and
maps di (and si in a weak simplicial case) are giving glueing instruction. Precise description (following [Lod-1]) is given below. The similar
construction for a cubic (or pre-cubic) set is described at the end of the
section30. We speculate also what should be a natural generalization
of (pre)simplicial and (pre)cubic categories.
30The
definition comes under the general scheme of a co-end, p. 371 of [BHS].
Knots and distributive homology
54
Let X be a simplicial space (e.g. simplicial set with discrete topology), and Y a cosimplicial space (e.g. N). We define their product over
∆ similar to the tensor product as follows:
[
X ×∆ Y =
(Xn × Yn )/ ∼rel
n≥0
where ∼rel is an equivalence relation generated by (x, fY (y)) ∼rel (fX (x), y),
f : [m] → [n], fX is the image of f under contravariant functor ∆ → X
and fY is the image of f under the covariant functor ∆ → Y.
Definition 13.1. The geometric realization of a simplicial space X is,
by definition, the space
[
(Xn × ∆n )/ ∼rel
|X | = X ×∆ N =
n≥0
We restrict our topological spaces in order to have |X × Z| = |X | × |Z|.
We can perform our construction also for a presimplicial set (or
space). The gluing maps are then limited to that induced by di .
If X is a simplicial set we may consider only nondegenerate elements
in Xn (that is elements which are not images under degeneracy maps)
and build |X | as a CW complex: We start from the union of n-cells ∆n
indexed by nondegenerate elements in Xn ; the face operations tells us
how these cells are glued together to form |X |.
Example 13.2. Let X be an abstract simplicial complex X = (V, P ).
If we order its vertices then X is a presimplicial set with Xn being
the set of n simplexes of X and face are defined on each simplex in a
standard way di (x0 , ..., xn ) = (x0 , ..., xi−1 , xi+1 , ..., xn A copresimplicial
space isPhere the category N with objects Yn = ∆n = {(y0 , ..., yn ) ∈
Rn+1 | ni=0 yi = 1, yi ≥ 0}, one object for any n ≥ 0. The coface maps
di : Yn → Yn+1 are given by di (y0 , .., yn ) = (y0 , ..., yi−1, 0, yi , ..., yn ) (of
course dj di = di dj−1 for i < j). Then the topological realization |X | is
a standard geometric simplicial complex associated to X , that is
[
|X| =
(Xn × ∆n )/(x, di (y) = (di (x), y) for x ∈ Xn and y ∈ ∆n−1 ,
n≥0
Xn with discrete topology and |X| with quotient topology.
13.1. Geometric realization of a (pre)cubic set. First we define
a precubic category ✷pre .
Definition 13.3. the precubic category ✷pre has as objects non-negative
integers [n] interpreted as n points ([0] has the empty object), see Figure
13.1. Thus the objects are the same as in presimplicial ∆pre category
except the grading shift (now [n] is in grading n).
55
[1])
[2])
[3]
[4])
Figure 13.1; interpreting objects [1], [2], [3] and [4] in ✷pre category
Morphisms are strictly increasing maps that is f ∈ Mor([m], [n]) if
f : (1, 2, ..., m) → (1, 2, ..., n) and i implies f (i) < f (j) with an additional data that points which are not in the image of f have 0 or 1
associated to them. Morphisms are generated by maps diǫ (1, 2, ..., n) =
(1, ..., i − 1, i + 1, ..., n) and the point i has marker ǫ (on the picture
marker 0 is denoted by ←) (here 1 ≤ i ≤ n + 1, ǫ = 0 or 1); see Figure
13.2.
The presimplicial category ∆pre is the quotient of the pre-cubic category ✷pre , but this functor is not that interesting in applications. The
proper functor is related with triangulation of a cube [Cla].
i=3
morphisms are
going up
d3
(0)
Figure 13.2; morphism (co-face map) d3(0) from [3] to [4]
Definition 13.4.
(i) A pre-cubic category is a contravariant functor from a pre-cubic category ✷pre to a given category, C, or,
equivalently, a covariant functor F : ✷op
pre → C. We can also say
that a pre-cubic category is a sequence of objects Xn in Ob(C),
and morphisms dǫi : Xn → Xn−1 satisfying dαi dβj = dβj−1 dαi for
i < j. The category ✷op
pre is visualized by the same diagrams as
the category ✷pre except morphisms will be read from the top to
the bottom.
56
Knots and distributive homology
(ii) A co-pre-cubic category is a covariant functor from a pre-cubic
category ✷pre to a given category, C. A basic example of a copre-cubic space (C = T OP ) is given by choosing X n = I n =
{(x1 , ..., xn ) ∈ Rn | 0 ≤ xi ≤ 1} and morphisms: diǫ (x1 , .., xn ) =
(x1 , ..., xi−1 , ǫ, xi+1 , ..., xn ). Let us denote this co-pre-cubic space
by pre .
The basic co-pre-cubic space, pre , can be enriched by degenerate
(projection) morphisms si : [n] → [n − 1] for 1 ≤ i ≤ n, given by
si (x1 , ..., xn ) = (x1 , ..., xi−1 , xi+1 , ..., xn ). We denote the new category
by . Then the morphisms diǫ and si (with the domain Xn = I n )
satisfy:
(1) djβ diα = diα dβj−1 for i < j,
(2) sj si = si sj+1 for i ≤ j,
diǫ sj−1 if i < j
(3) sj diǫ =
j
di−1
if i > j
ǫ s
i i
(4) s dǫ = IdI n .
The axioms of the cubic category are modeled on axioms (1)-(4)
more precisely:
Definition 13.5.
(i) The cubic category ✷ is composed of objects
[n], face maps diǫ (as in pre-cubic category ✷pre ) and degeneracy
maps si : [n − 1] → [n] (1 ≤ i ≤ n and relations between
morphisms in the category are given by axioms (1)-(4).
(ii) The cubic category is a functor F : ✷op → C. The classical
example (of a cubic space) is giving by an approach to singular
cubic homology. Here, for a topological space T , F ([n]) is the
set of all continuous maps f : I n → T . dǫi : F ([n]) → F ([n − 1])
di
f
is given by dǫi (f ) = f diǫ : I n−1 →ǫ I n → T .
(iii) The co-cubic category is a functor F : ✷ → C. The classical
example (I n , diǫ , si ) was described before as a motivation for a
cubic category, . This example is used as a building block of
a geometric realization of a pre-cubic and cubic set (Definitions
13.6, 13.8).
As we observed already, the pre-cubic category leads to two presim(0)
1)
plicial categories (Xn , di ) and (Xn , di (with shifted grading). Conversely, if we have two presimplicial categories so that dαi dβj = dβj−1dαi
then we combine them to in pre-cubic category (with grade shift).
With degenerate maps situation is not that clear as a condition (3)(4) of a simplicial category only partially agree with the analogous
conditions of a cubic category.
57
The geometrical realization of a cubical and precubical set (or space)
is analogous to that for simplicial or presimplicial sets (space). We
write below a formal definition only in the case of a pre-cubic set (space)
as degeneracies of a cubic set are not necessarily the one used in knot
theory.
Definition 13.6. The geometric realization of a precubic set is a CW
complex defined as follows (notice that Xn is indexing cubes and precubic structure gives an instruction how to glue the cubes together):
[
(Xn × I n )/ ∼rel
|X | = X ×✷pre =
n≥0
where ∼rel is an equivalence relation generated by (x, diǫ (y)) ∼rel (dǫi (x), y),
and, as before diǫ : I n−1 → I n and dǫi : Xi → Xi−1 , x ∈ Xn , and
y ∈ I n−1 .
More generally:
Definition 13.7. Let X be a pre-cubic space (e.g. pre-cubic set with
discrete topology), and Y a co-pre-cubic space (e.g. pre ). The we
define
[
(Xn × Yn )/ ∼rel .
X ×✷pre Y =
n≥0
where ∼rel is an equivalence relation generated by (x, diǫ (y)) ∼rel (dǫi (x), y).
If (Xn , dǫi , si ) is a cubic set (or space) we define its geometric realization similarly to Definitions 13.6 and 13.7, but including also si , si
as gluing morphisms (effectively dividing by degenerate part, which is
not necessary acyclic).
Definition 13.8. Let X be a cubic set and Y the co-cubic space (I n , diǫ , si )
then we define the geometric realization |X | of X as
[
|X | = X ×✷ =
(Xn × I n )/ ∼rel
n≥0
where ∼rel is an equivalence relation generated by (x, diǫ (y)) ∼rel (dǫi (x), y),
and analogously with si and si .
14. Higher dimensional knot theory M n → Rn+2
Many ideas described in this paper can be applied to higher dimensional knot theory, were we study embeddings of n dimensional (mostly
orientable) manifolds in Rn+2 . For this we direct readers to [CKS] and
[Pr-Ro-2].
58
Knots and distributive homology
15. Acknowledgements
J. H. Przytycki was partially supported by the NSA-AMS 091111
grant, by the GWU-REF grant, and Simons Collaboration Grant316446.
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Department of Mathematics,
The George Washington University,
Washington, DC 20052
e-mail: [email protected],
University of Maryland College Park,
and University of Gdańsk, Poland